sustainability
Article
Optimal Scheduling of Rainwater Collection Vehicles: Mixed
Integer Programming and Genetic Algorithms
Mohammed Alnahhal 1, * , Nikola Gjeldum 2 and Bashir Salah 3
1
2
3
*
Citation: Alnahhal, M.; Gjeldum, N.;
Salah, B. Optimal Scheduling of
Rainwater Collection Vehicles: Mixed
Integer Programming and Genetic
Algorithms. Sustainability 2023, 15,
Mechanical and Industrial Engineering Department, American University of Ras Al Khaimah,
Ras Al Khaimah P.O. Box 10021, United Arab Emirates
Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture (FESB), University of Split,
21000 Split, Croatia; ngjeldum@fesb.hr
Industrial Engineering Department, College of Engineering, King Saud University, P.O. Box 800,
Riyadh 11421, Saudi Arabia; bsalah@ksu.edu.sa
Correspondence: mohammed.alnahhal@aurak.ac.ae
Abstract: Due to climate change, some areas in the world witnessed higher levels of heavy rain
than the capacity of the wastewater system of the streets. Therefore, water tankers are used for the
dewatering process to take the extra rainwater from the streets to keep a smooth flow of vehicles and
to use the water in agriculture and industry. Water is taken to a water treatment plant. Performing
the dewatering process as fast as possible, especially in crowded streets, was ignored by researchers.
In this study, at first, the problem was solved using two mixed integer programming (MIP) models.
A new variant of identical parallel machine scheduling with job splitting is proposed for the first
time, where one or at most two tankers can work at the same flood location at the same time. This
is performed in the second model. However, the first model considers dividing the dewatering
processes into two phases, where the first one, which is more urgent, is to reduce the amount of
floodwater. The second one is for dewatering the rest of the water. Then two genetic algorithms
(GAs) were used to solve faster the two MIP models, which are NP-hard problems. At first, the MIP
and GA models were applied to small-sized problems. Then GA was used for large practical data
sets. Results showed that for small problems, MIP and GA gave optimal solutions in a reasonable
number of iterations, while for larger problems, good solutions were obtained in a reasonable number
of iterations.
Keywords: water tankers; water treatment plant; flood; parallel machines scheduling; mixed integer
programming; genetic algorithm
9252. https://doi.org/10.3390/
su15129252
Academic Editors: Eirini Aivazidou
and Naoum Tsolakis
Received: 30 April 2023
Revised: 23 May 2023
Accepted: 6 June 2023
Published: 8 June 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1. Introduction
Because of climate change, some areas face rain levels more than expected. The
infrastructure of some cities is not capable of dealing with such huge amounts of rainwater [1]. Therefore, water tankers are used in the winter to remove extra rainwater from the
streets. The study focuses on a certain city that faces levels of rainwater that need to be
evacuated from the streets to a water treatment plant in the city. Usually, water tankers
offer a mobile solution to several bulk water needs. The problem of allocating different
trucks to different water locations and the optimal scheduling of trucks was ignored in
previous studies. Therefore, this study aims to fill in this gap of research. There are some
similarities between the traditional job scheduling problem and the water tankers problem
(WTP). However, there are some special conditions of WTP that make it different and need
further investigation.
The problem investigated in this study is similar to parallel machine scheduling with a
job-splitting property (PMSJS) [2]. The machines here are water tankers. There are no setup
times. The jobs are dewatering processes of flood spots on the main streets in a city. The
main difference between PMSJS and WTP is the maximum job-splitting of only 2. In other
Sustainability 2023, 15, 9252. https://doi.org/10.3390/su15129252
https://www.mdpi.com/journal/sustainability
Sustainability 2023, 15, 9252
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words, two trucks at maximum can work at the same flood location from two sides of the
pool of water. To solve the problem, mixed integer programming (MIP) is used. Because
it is NP-hard, a genetic algorithm (GA) is used to solve it. Two models are created, and
each model adds some of the practical considerations to the problem in order to arrive at a
solution. These models are as follows:
1.
2.
Model 1: several trucks work together in the same area or city, but every location can
be dewatered by only one truck. The problem combines parallel machine scheduling
with job shop scheduling. In this case, some locations exist on very active streets, and
flood level matters. Therefore, two phases for such a location are distinguished. The
first phase of the deep flood, which is more important, must be performed before the
second phase. Unlike job shop scheduling problems, parallel machine scheduling
problems have no predefined routes for jobs on the machines. There is a single
production stage for each job [3].
Model 2: it is similar to parallel machine scheduling with a job-splitting property but
with a special condition of two water tankers per location at maximum.
The objective will be to minimize the total weighted flow time. None of the previous
studies investigated water tanker allocation and scheduling problems. Therefore, this
study will investigate that problem. However, the study did not examine combining
models 1 and 2, which is interesting for future research. This study is based on a practical
case study in which digital tools used in Industry 4.0, such as sensors to measure the level
of water, are not used because of budget limitations. The aim of this study is to provide
algorithms to optimize the system based on the current conditions. However, future work
can investigate the effect of such tools on the optimization methods and improve results.
The remainder of this paper is structured as follows: a literature review of different types
of machine scheduling is presented in the next section. Then, in Section 3, the methodology
containing the two MIP and GA models is explained. In Section 4, the results and analysis
are presented. Finally, conclusions, including study limitations and recommendations for
future research, are presented.
2. Literature Review
There are several methods of rainwater management used in the world and investigated in the literature, such as retention reservoirs, infiltration boxes, and green roofs.
For example, the benefits of a green roof test bed for stormwater management were examined in a study by Qin et al. [4]. Such systems were found to be useful in tropical areas.
Moreover, Eger et al. [5] asked how can the hydrologic processes influence stormwater
infrastructure behavior. Zhang et al. [6] compared stormwater management in two cities:
Singapore and Berlin. Different solutions for stormwater management were provided
for the two cities. A cost analysis was provided. It was found that green roofs result in
energy savings. Furthermore, various sustainable stormwater management solutions were
evaluated by Alkhaledi et al. [7] using a multi-criteria decision-making model. The purpose
is to guarantee public safety and adhere to runoff regulations.
The scheduling problem was investigated in environments other than transportation,
such as the production process. For example, Shevasuthisilp and Intawong [8] investigated
the scheduling problem in production planning to reduce makespan error by suggesting the
best sequencing process to reduce the total processing time. Moreover, Dupláková et al. [9]
investigated the scheduling problem in the production process in which the optimal sequencing of parts to be machined is determined. They used different methods, such as
simulation. In another study by Balog et al. [10], the scheduling of manufacturing was
investigated using Scheduling Software Lekin.
There are no publications about water tanker optimal schedules. There was a literature
review that considered other problems. Stolk et al. [11] investigated water “tank” delivery
schedules in Australia, which differs from water “tankers” in our study. The objective is
to minimize the cost of delivering a given amount of sales value. Tankers in other fields,
such as the petroleum and chemical industries, were investigated. Some studies considered
Sustainability 2023, 15, 9252
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coastal shuttle tanker scheduling, such as the study by Yang et al. [12]. They investigated
the tanker scheduling plan model and considered the tanker fleet design. Cankaya et al. [13]
investigated the scheduling of chemical tankers.
The water tanker scheduling was not considered before. This study presents two
models. They are more complicated than the single machine sequencing model which was
considered in the study by Baker and Keller [14], whey they provided comparisons between
several integer programming models. This study considers several trucks working in the
same area. In many cases, the sequencing problem can be modeled as a traveling salesman
problem (TSP). Some studies investigated multiple TSP where some salesmen visit all the
cities [15]. This can be helpful for real-life applications [16]. In the first model of this study,
two phases exist, where the first phase must be finished before the start of the second one.
Generally, when tasks depend on each other, job shop scheduling is utilized. Job shop
scheduling was used in several studies, such as with Bülbül and Kaminsky [17], who made
a new heuristic for large practical problems. Moreover, Ku and Beck [18] evaluated four
MIP models of traditional job shop problems.
However, our study is not exactly a job shop problem because it has its unique conditions. This study is more similar to the identical parallel-machine scheduling but with zero
setup time. Lee and Pinedo [19] considered identical machines in parallel and considered
due dates. They took into consideration the setup time. Fanjul-Peyro et al. [20] used MIP to
solve the unrelated parallel machine problem. In unrelated machines scheduling, the time
needed for a given job depends on the machine. Tavakoli and Mahdizadeh [21] presented
an integer-linear programming model for an identical parallel-machine scheduling problem.
They proposed GA to solve the problem for large data sets. The objective was to minimize
the total weighted flow time. The principles of natural evolution are the basis for GAs,
which are search algorithms. GAs are useful for optimization problems, where the goal is to
find the best solution out of a set of possible solutions. They are also applicable to problems
that do not have a known solution, such as prediction problems. GAs can be employed
to solve problems that are too complex for traditional algorithms [22]. A GA can evolve
remarkably complex and interesting structures despite its highly simplified computational
setting [23]. In GA, it is very important to define the input parameters, such as the crossover
point [24]. Researchers can test different types of representations, crossover and mutation
operators, and different methods of reproduction and selection. However, all these methods
are inspired by Holland’s original GA and biological evolution [23]. Another study that
used GA in identical parallel machine scheduling is the one by Chaudhry and Drake [25],
who took into consideration the workers’ assignments. Moreover, Demirel et al. [26] also
used GA in parallel machine scheduling to minimize total tardiness.
On the other hand, Dell’Amico et al. [27] presented heuristic and exact algorithms for
such problems. The same problem was considered in the study by Fleszar and Hindi [28],
who considered the resource constraint. The secondary resource constraint was also
studied by Vallada et al. [29], who used enriched metaheuristics to solve the problem.
Furthermore, Abreu and Prata [30] used a hybrid meta-heuristic to solve the scheduling
problem. Simulated annealing was presented for unrelated parallel machine scheduling
by Jouhari et al. [31]. Furthermore, Yepes-Borrero et al. [32] presented two metaheuristics
to solve the unrelated parallel machine scheduling. Fanjul-Peyro [33] investigated the
problem where processing times and setups depend on both the machine and the job. They
presented MIP and a three-phase algorithm to solve the problem. Alharkan et al. [34]
presented Tabu search and particle swarm optimization algorithms to solve the scheduling
of two parallel machines with a single server.
However, in parallel machine scheduling problems, each job is assigned to exactly
one machine. In our study, however, it can be assigned to two trucks at most in model 2.
Therefore, a new variant, which is job splitting, is needed. Kim et al. [2] considered the
problem where several sub-jobs could be processed separately on parallel machines by
splitting a job. They suggested a two-phase heuristic algorithm. Sarıçiçek and Çelik, C. [35]
investigated the parallel machine scheduling problem with job splitting where the objective
Sustainability 2023, 15, 9252
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is to reduce tardiness. Eroğlu et al. [36] solved the unrelated machine scheduling problem
with job splitting using GA with the objective of reducing the makespan (completion time).
They considered a real-life problem where each machine has its own processing time. Lee
and Jang [3] investigated machine scheduling with job splitting and dedicated machines.
Dedicated machines mean that jobs can only be processed on a specified set of machines.
They assumed sequence-independent setup times. Oktafiani and Ardiansyah [37] investigated the scheduling problem of identical parallel machines with job splitting using MIP,
where the objective is to reduce the makespan.
Rainwater collection optimization was overlooked in the literature. Moreover, most
of the above studies considered makespan or total tardiness as objectives. The main
contribution of this study is to investigate an identical parallel machine with no setup time
and with the objective of minimizing the total weighted flow time with job splitting in a
real-life application where job splitting is limited to two machines (water tankers). Such
a real-life application was ignored in the literature. Furthermore, the division of parallel
machine scheduling into two phases is also new.
3. Materials and Methods
The methodology of rainwater management proposed in this study was based on a
real case study in the Middle East. Many projects of rainwater management can be strategic
ones that need major changes in infrastructure and need large investment costs for a long
period. However, the flood can come suddenly and needs immediate solutions on the
operational level. This study investigates such an operational intervention. However, the
main weakness of water tankers is that the needed time for the dewatering process using
water tankers is much longer than what other methods need. The study tries to reduce
that time by providing mathematical modeling that gives solutions in a reasonable time.
Therefore, when enough fund is available in later stages, changes in the infrastructure
are required. This paper proposes a customized way of GA that is suitable for identical
parallel machine scheduling with job splitting where the maximum number of assigned
machines for a job is two tankers. This new variant of identical parallel machine scheduling
is proposed for the first time.
The methodology depends on two MIP and GA models. The following assumptions
were considered in this study:
•
•
•
•
•
•
All the trucks are with the same capacity and the same loading and unloading rate.
Therefore, the machines (trucks) are identical.
For the first model, one truck is only capable of dewatering one location at a time. In
the second model, two trucks can perform the job.
Based on experts’ opinions and historical data, the exact number of truckloads of a
certain water location is known with certainty.
All the water will be taken to the same final location (the water treatment plant in the city).
The penalty exists until the whole flood location is empty of water and the truck
unloaded the last truckload in the water treatment plant.
The available number of trucks is predetermined. Therefore, it is not a decision variable.
WTP has unique conditions and, therefore, the following considerations need to be
taken into account:
•
•
•
•
•
•
Different locations have different traffic volumes, and therefore, they have different priorities.
The dewatering process at some locations can be divided into two phases (deep water
and shallow water). Different priorities are assigned to different phases. In this case,
phase 1 must be done before Phase 2.
There is no relationship between one location and another, and this means that the
problem is not like the job shop scheduling, except for the two phases of model 1.
No setup time is considered.
In the second model, there is no idle time.
The objective is to minimize total weighted flow time.

Sustainability 2023, 15, 9252



and shallow water). Different priorities are assigned to different phases. In this case,
phase 1 must be done before Phase 2.
There is no relationship between one location and another, and this means that the
problem is not like the job shop scheduling, except for the two phases of model 1.
No setup time is considered.
5 of 18
In the second model, there is no idle time.
The objective is to minimize total weighted flow time.
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1. Study
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is considered
a decision
variable.
Some
locations
can
have massive amounts of rainwater on important streets with large traffic volumes. In
such locations, the first phase of rainwater removal is more important than the second
phase. The first phase must be done before the second one. Each phase of each location
must be performed by the same truck. Each location does not need to have two phases,
because some streets are not with heavy traffic or because the level of rainwater is shallow.
Table 1 shows an example of six flood locations. The time unit is the time needed for
the dewatering of one truckload. Each location needs a different number of dewatering
processes (truckloads). The dewatering process means sucking the rainwater from the
Sustainability 2023, 15, 9252
6 of 18
street into the truck and unloading rainwater from the truck into the water treatment plant.
If it is performed five times, then d = 5. For example, the truck must work on the first
location for dewatering five times. Each time, transportation occurs. Therefore, the total
time spent serving location 1 is 10 units of time which are (1 + t) d = 5 × (1 + 1). The unit of
time is the average needed time to finish one truckload, including loading and unloading
of rainwater, without considering transportation. The traffic volume is measured on a
scale of 1 to 10, where 10 is very heavy traffic in the middle of the city, and 1 means very
distant areas with lower importance. Therefore, the model gives more priority to areas in
the middle of the city with p-values approaching 10.
Table 1. Data of Model 1.
Flood Location
Truckloads (d)
Transportation to and from Plant (t)
Street Traffic (p)
Deep/Shallow δ(β)
1
2
3
4
5
6
5
10
15
7
10
8
1
0.5
0.75
0.5
0.7
1
5
7
6
4
3
8
1
2(3) + 1(7)
2(7) + 1(8)
1
1
1
Table 1 shows some locations with massive amounts of water (e.g., locations 2 and 3).
The importance of a location depends on how deep the water is. A deeper flood has higher
importance. Therefore, the same location can be divided into two locations. The first phase
of dewatering is more important. Assume there are two levels (deep and shallow). δ = 1
means shallow and δ = 2 means deep. The expression δ(β) = 2(3) means that the first three
truckloads are in the first phase. For the second flood location, the first three truckloads are
considered more important than the last seven truckloads in Phase 2. Therefore, Phase 1
has a higher p-value, for example (8), to increase the attention for these phases. The first
phase (e.g., three truckloads in the second location) must be finished before the rest of
the processes are complete. The two phases can be accomplished by two different trucks.
Therefore, Table 1 can be rewritten as in Table 2.
Table 2. A modified table for Model 1.
Real Flood Location
1
2
3
4
5
6
Model Flood
Location
Truckloads (d)
Transportation to
and from Plant (t)
Street Traffic (p)
1
2
3
4
5
6
7
8
5
3
7
7
8
7
10
8
1
0.5
0.5
0.75
0.75
0.5
0.7
1
5
7+1=8
7
6+2=8
6
4
3
8
Previous Model
Flood Location (ν)
2
4
Mathematical modeling will depend on Table 2, assuming that we have eight locations
instead of six.
The due date is zero, which means that trucks must start immediately, and therefore,
there is no holding cost. Holding cost occurs if the process is done too early. Loading and
unloading needed times of water are assumed fixed. The decision variables of the study are:
xj is the start time (in dewatering time units) for flood location j (measured from time
zero). For example, x1 = 0 means that the truck will start from the flood location 1.
wmj =
1, i f truck m is assigned to location j
0, otherwise
Sustainability 2023, 15, 9252
7 of 18
ymij =
1, i f i precedes j using truck m
0, i f j precedes i using truck m
T, which is an input parameter, is the total number of water tankers. The model can be
written as follows:
For example, y112 means that truck 1 will start serving location 1 before location 2.
However, it does not mean that location 2 will immediately be after location 1.
The model can be written as follows:
N
minimize z =
∑ p j s+j
(1)
j =1
Subject To
xi − x j + Mymij − M wmi + wmj − 2 ≥ 1 + t j d j + M wmi + wmj − 2
∀m = 1 . . . T, ∀i
= 1 . . . N − 1, ∀ j = i + 1 . . . N
− xi + x j − Mymij − M wmi + wmj − 2 ≥ (1 + ti )di − M + M wmi + wmj − 2
∀m = 1 . . . T, ∀i
= 1 . . . N − 1, ∀ j = i + 1 . . . N
+
x j + s−
∀j = 1 . . . N
j − sj = − 1 + tj dj
(2)
(3)
(4)
M
∑
∀m = 1 . . . T, ∀ j = 1 . . . N
wmj = 1
(5)
m =1
− x i + x i +1 ≥ (1 + t i ) d i
−
x j , s+
j , sj ≥ 0
ymij and wmj = (0, 1)
∀i ∈ ν
∀j = 1 . . . N
∀m = 1 . . . T, ∀i = 1 . . . N, ∀ j = 1 . . . N, i 6= j
(6)
(7)
(8)
where M is a large number. The objective function is the total weighted flow time for
different locations. The flow time of a certain location means the time needed to finish all
the truckloads of that location. In other words, it is the end time of that location. To find the
objective function, the end time for each location is multiplied by the level of street traffic
(p). Then the summation for all locations is found. Makespan was not considered as the
objective because different flood locations have different levels of importance. s+
j is used to
represent the end time of task j. This means that there is a penalty until the location is empty
of water. Equations (2) and (3) are to guarantee that two locations
cannot be served by
one truck at the same time. The two terms − M wmi + wmj − 2 and + M wmi + wmj − 2
are added to these two constraints to enforce the model to consider these two constraints
if the term is zero. In other words, if the locations i and j are served by the same truck,
m, then they cannot be served at the same time. If one or the two
of these locations are
not served by the truck m, then the term
−
M
w
+
w
−
2
will be a large positive
mi
mj
number, and the term M wmi + wmj − 2 be a large negative number, and therefore, the
two constraints (2) and (3) will be redundant for the truck m. Equation (4) is to calculate
the end time of dewatering. The end time here means that the truck made both the loading
and unloading of water. Constraint (5) is added because we have only one truck for one
location. Constraint (6) was added to guarantee that the first phase was finished before
the second one. Equation (7) is the nonnegative constraint for the continuous variables
−
x j , s+
j , s j . The last Equation (8) is for ymij and wmj , which is binary.
MIP for Model 2
This model represents an innovative variant of the job-sequencing model where the
same job (dewatering of flood location) can be performed by one or two trucks simultane-
Sustainability 2023, 15, 9252
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ously or at different times. Two trucks on the two sides of the flood location can exist. It
is not necessary that they work exactly at the same time. However, it is possible. In other
words, the two trucks can work at the same time or at different times. This is important for
some locations that are important in the middle of the city, assuming that there is enough
space for two trucks. This is the main difference between the traditional parallel machine
scheduling and WTP defined in this paper.
The following two variables are introduced:
•
•
Amj number of truckloads at location j performed by truck m.
Moreover, xmj is used to represent the starting time in which truck m works on location j.
The model can be written as follows:
N
Minimize z =
∑ p j s+j
(9)
j =1
Subject to the following:
xmi − xmj + Mymij − M wmi + wmj − 2 ≥ 1 + t j Amj + M wmi + wmj − 2
= 1 . . . N − 1, ∀ j = i + 1 . . . N
− x mi + xmj − Mymij − M wmi + wmj − 2 ≥ (1 + ti ) Ami − M + M wmi + wmj − 2
= 1 . . . N − 1, ∀ j = i + 1 . . . N
xmj + 1 + t j Amj ≤ s+
j
∀m = 1 . . . T, ∀i
∀m = 1 . . . T, ∀i
∀m = 1 . . . T, ∀ j = 1 . . . N
(10)
(11)
(12)
M
∑
wmj ≤ 2
∀j = 1 . . . N
(13)
m =1
T
∑
Amj ≥ d j
∀j = 1 . . . N
(14)
Amj ≤ Mwmj
∀m = 1 . . . T, ∀ j = 1 . . . N
(15)
Amj is integer
∀m = 1 . . . T, ∀ j = 1 . . . N
(16)
∀j = 1 . . . N
(17)
m =1
−
x j , s+
j , sj ≥ 0
yij = (0, 1)
wmj = c(0, 1)
∀i = 1 . . . N, ∀ j = 1 . . . N, i 6= j
∀m = 1 . . . T, ∀ j = 1 . . . N
(18)
(19)
The first two constraints (Equations (10) and (11)) contain Amj instead of dj , and xmi
and xmj instead of xi and xj in the previous model. This is because a truck m can take
some or all the demand for the location j. It is possible; however, it takes no jobs, and the
other trucks can do the job. Constraint (12) is to find the end time of truck m on location j.
Constraint (13) is to let two trucks at maximum work in the same location. Constraint (14)
is to guarantee that all the truckloads served by all the trucks are exactly as the truckload
found in the street (dj ). Constraint (15) is to enforce the model to assign 1 for wmj whenever
Amj is greater than zero. This model contains all types of variables: continuous, integer, and
binary, as shown in the last four constraints.
Genetic Algorithms
GA was used to solve the two MIP models in reasonable times. There are some steps
of GA, such as selecting input parameters such as population size and mutation rate, chroSustainability 2023, 15, 9252
9 of 18
mosome formulations, mating process, and mutation process. Mating and mutation processes are repeated for a certain number of iterations or when a certain criterion is reached.
Genetic
Algorithms
In this study, a certain
number
of iterations is predefined, such as 100 iterations, so that
GA was used toThe
solvereader
the two MIP
models
in reasonable
times.
are some
processing time is still reasonable.
might
refer
to Haupt
andThere
Haupt
[38]steps
for
of GA, such as selecting input parameters such as population size and mutation rate,
more information about
GA. The main contribution of this study about GA is the formuchromosome formulations, mating process, and mutation process. Mating and mutation
lation of chromosomes
that are
take
into consideration
theofsequence
ofwhen
jobsaand
thecriterion
assignprocesses
repeated
for a certain number
iterations or
certain
is
reached. In
study, a certain
number
of iterations
is predefined,
100 iterations,
ment of each job (location)
tothis
different
tankers.
Two
GA models
were such
usedasfor
the two
so that processing time is still reasonable. The reader might refer to Haupt and Haupt [38]
models.
GA for Model 1
for more information about GA. The main contribution of this study about GA is the
formulation of chromosomes that take into consideration the sequence of jobs and the
assignment of each job (location) to different tankers. Two GA models were used for the
the
phases of some flood locations are used. The initial chromotwo two
models.
In this model,
somes are built. Then,
“repair”
GAafor
Model 1 function is needed if the second phase comes before the
first one. The allocation of
themodel,
trucksthetotwo
different
exist
theThe
chromosomes.
In this
phases oflocations
some floodmust
locations
arein
used.
initial chromoEach gene was chosen
to are
contain
two anumbers:
the sequence
the truck.
We call
each
somes
built. Then,
“repair” function
is needed if and
the second
phase comes
before
the
first
one.
The
allocation
of
the
trucks
to
different
locations
must
exist
in
the
chromosomes.
one a sub-gene. Figure 2 shows how to interpret the chromosome for a certain parent (P1).
Each gene was chosen to contain two numbers: the sequence and the truck. We call each
Note that we can putone
thea genes
of each truck beside each other. It will not affect the meansub-gene. Figure 2 shows how to interpret the chromosome for a certain parent
ing or the objective function
value.
(P1). Note that
we can put the genes of each truck beside each other. It will not affect the
meaning or the objective function value.
Figure
2. Chromosomes
Figure 2. Chromosomes
meaning
in GA. meaning in GA.
For mating, the whole gene containing the sequence and the truck must stay in one
Forcontaining
example, if the
crossover
pointand
is 3, then
firstmust
two genes
subFor mating, thechromosome.
whole gene
the
sequence
the the
truck
stay(four
in one
genes) must be exchanged between the parents. For mutation, three different random
chromosome. For example, if the crossover point is 3, then the first two genes (four subnumbers are generated: one for rows and two for columns. Then, a fourth random number
genes) must be exchanged
between
the choose
parents.
mutation,
threewill
different
random
is generated
to randomly
if theFor
sequence
or the trucks
be changed.
If the
random
is greater
sequence needs
changed,
and the two
genes
numbers are generated:
onenumber
for rows
and than
two0.5,
forthe
columns.
Then,toabefourth
random
number
must exchange their sequence. If it is less than 0.5, the truck numbers need to be exchanged.
is generated to randomly choose if the sequence or the trucks will be changed. If the ranFigure 3 shows the mating and mutation. In the mutation of C1, the truck numbers of the
dom number is greater
than
0.5,fourth
the genes
sequence
needs to be changed, and the two genes
second
and the
were exchanged.
Sustainability 2023, 15, x FOR PEERmust
REVIEW
exchange
Sustainability 2023, 15, 9252
10 of need
19
their sequence. If it is less than 0.5, the truck numbers
to b
changed. Figure 3 shows the mating and mutation. In the mutation of C1,
the truck
10 of 18
bers of the second and the fourth genes were exchanged.
must exchange their sequence. If it is less than 0.5, the truck numbers need to be exchanged. Figure 3 shows the mating and mutation. In the mutation of C1, the truck numbers of the second and the fourth genes were exchanged.
Figure 3. Mating and mutation in model 1.
Figure3.3.Mating
Mating and
in model
1.
Figure
andmutation
mutation
in model
1.
However, Table 2 contains eight flood locations to account for the two phases. Before
However, Table 2 contains eight flood locations to account for the two phases. Before
making
the above
mating
and mutation,
function istoneeded.
Figure
shows
However,
Table
2 contains
eightaa “repair”
flood
locations
account
for 4the
twothe
phases. B
making the above
mating
and mutation,
“repair”
function is needed.
Figure
4 shows
the
“repair”
function.
For
the
same
truck
2,
location
2
(phase
1)
must
be
served
before
location
making
aboveFor
mating
and
mutation,
a 2“repair”
is needed.
Figure 4 show
“repair”the
function.
the same
truck
2, location
(phase 1) function
must be served
before location
3 (phase 2).
3 (phasefunction.
2).
“repair”
For the same truck 2, location 2 (phase 1) must be served before loc
3 (phase 2).
Figure 4. Repaired chromosome in model 1.
Figure 4. Repaired chromosome in model 1.
For the following chromosome, the initial calculations of the costs are shown in Table 3.
For the
following
chromosome,
calculations
of the
costs
are phase
shown1in(3Table
However,
these
calculations
neglect the
the initial
fact that
phase 2 must
come
after
must
3.
However,
these
calculations
neglect
the
fact that some
phaseadjustments
2 must comemust
afterbephase
1 for
(3
come
after
2,
and
5
must
come
after
4).
Therefore,
made
Figure 4. Repaired chromosome in model 1.
must
come
after
2,
and
5
must
come
after
4).
Therefore,
some
adjustments
must
be
made
these four locations, as in Table 4.
for these four locations, as in Table 4.
5For the
1
following
chromosome,
the
initial
of4 the costs
in
6
2
1
1
3
1
2
2calculations
8
2
2
7are shown
1
3. However,
these
neglect
24 must
after
phas
5
1
6
2 calculations
1
1
3
1 the
2 fact
2 that
8 phase
2
2 come
7
1
Table 3. Initial cost calculation in Model 1.
must come after 2, and 5 must come after 4). Therefore, some adjustments must be
3. Initial cost calculation in Model 1.
forTable
these
four locations,
ast in Table 4.p
d
(1 + t)d
End Time (s+ )
ps+
d
t
p
Truck 1
5
8
0.75
Truck
11
5
6
2
1
1
36
1
5
1
5
8
0.75
65
3
7
0.5
7
1
5
1
5
10
0.7
Table 3.7Initial cost
calculation
in Model31.
32
7
0.5
7
Truck
7d
0.5
76
10
0.7 t
34
2
3
0.5
8
Truck 2
Truck
18
8
1
8
6
7
0.5
4
4
7
0.75
8
5
8
0.75
2
18
3
7
Truck 2
6
2
3
85
7
10
7
3
0.5
1
1
0.5
0.7
0.5
0.5
8
8
(1 + t)d
1
p
6
5
7
3
4
8
14
2
1410
10.5
1017
End Time (s+)
2
10.5
10.5 (1
17
4.5
16
10.5
12.25
4.5
16
+ t)d
14
10
10.5
17
10.5
4.5
814
24
14
34.5
24
51.5
2
34.5
10.5 End
51.5
15
31
10.5
43.25
15 Sum
31
ps+
4 84
120
84
241.5
120
154.5
2
241.5
42(s+)
154.5
Time
120
248
42
346
14 120
1356
24 248
34.5
51.5
10.5
15
7
ps
84
12
241
154
42
12
4
7
0.75
8
12.25
43.25
346
1356
Sum
Table 4. The adjusted calculation for Table 3.
Truck
Location
d
t
p
(1 + t) d
Start Time (x)
The two phases’ locations
1
5
8
0.75
6
14
0
Table
4.
The
adjusted
calculation
for
Table
3.
1
3
7
0.5
7
10.5
24
2
2
3
0.5
8
4.5
10.5
Truck
Location
d
t
p
(1 + t) d
Start Time (x)
2
4
7
0.75
8
12.3
31
The two phases’ locations
Sorted1phases
5
8
0.75
6
14
0
0.5
8 7
4.5
10.5 24
12
3 2
7 3
0.5
10.5
21
2 3
3 7
0.5
4.5
10.5
0.5
7 8
10.5
24
22
4 4
7 7
0.75
8
12.3
31
0.75
8
12.3
31
Sorted phases
1
5
8
0.75
6
14
0
2
2
3
0.5
8
4.5
10.5
Adjusted
1 start and end
3 times
7
0.5
7
10.5
24
0.5
8 8
4.5
10.5 31
22
4 2
7 3
0.75
12.3
11
5 3
8 7
0.75
14
0
0.5
7 6
10.5
24
Adjusted start and end times
2
4
7
0.75
8
12.3
31
2
2
3
0.5
8
4.5
10.5
1
5
8
0.75
6
14
43.3
1
3
7
0.5
7
10.5
24
Sustainability 2023, 15, 9252
2
1
4
5
End Time (s11+)of 18
14
34.5
15
End Time (s+ )
43.3
14
15
34.5
15
34.5
43.3
43.3
14
15
34.5
15
43.3
14
34.5
43.3
15
57.3
34.5
7
0.75
8
12.3
31
43.3
8The extra 43.3
0.75 is multiplied
6 by p-value,14
43.3 total cost is 1356
57.3+ 259.5 =
which is 6. So the
1615.5.
GA for
2 is multiplied by p-value, which is 6. So the total cost is 1356 + 259.5 = 1615.5.
TheModel
extra 43.3
First truck Second Truck
First truck
Second Truck
In this model, one or two trucks can work on a location simultaneously or at different
GA for Model 2
times. In the previous model, we used two sub-genes inside each gene to represent a loIn sequence
this model,
or two
trucks
canfive
work
on a location
or at difcation
andone
truck.
In this
model,
sub-genes
will besimultaneously
used for each location
as
ferent
times. In the previous model, we used two sub-genes inside each gene to reprefollows:
sent a location sequence and truck. In this model, five sub-genes will be used for each
location
as follows:
% of truckload
performed by the
Sequence of the second
Sequence of the first truck
truckperformed by
% of first
truckload
Sequencetruck
of the second
the first truck
Sequence of the first truck
truck
To facilitate mating and mutation, the last two sub-genes will be kept as two random
To facilitate mating and mutation, the last two sub-genes will be kept as two random
variables between 0 and 1, and they are converted to integers in the “cost” function. Figure
variables between 0 and 1, and they are converted to integers in the “cost” function. Figure 5
5 shows 6-location chromosomes in population matrix form and in the cost model form.
shows 6-location chromosomes in population matrix form and in the cost model form.
The second and the third rows in the figure are used in the cost function after converting
The second and the third rows in the figure are used in the cost function after converting
numbers between 0 and 1 into integers. The third row has the sequence and the truckload
numbers between 0 and 1 into integers. The third row has the sequence and the truckload
per truck in integer numbers.
per truck in integer numbers.
Figure
Figure5.5.Chromosomes
Chromosomesrepresentations
representationsin
inmodel
model22in
inthe
thepopulation
populationmatrix
matrixand
andthe
thecost
costfunction.
function.
In
Inthe
the first
first row,
row,the
thesequence
sequenceof
ofthe
thefirst
firstand
andsecond
secondtrucks
trucksisis represented
representedas
asaarandom
random
number.
Then,
in
the
second
one,
the
random
number
is
converted
into
an
integer
to
number. Then, in the second one, the random number is converted into an integer
to reprepresent
the
sequence.
For
example,
for
truck
1,
the
minimum
random
number
was
resent the sequence. For example, for truck 1, the minimum random number was
0.01560744
0.01560744 (only
(only three
three digits
digits are
are shown
shown in
inFigure
Figure55as
aso.016).
o.016).Therefore,
Therefore,the
thefirst-served
first-served
location will be location 6. Even though the last sub-gene contains a different number
location will be location 6. Even though the last sub-gene contains a different number
(shown as 0.879), the gene is “repaired” so that if the two trucks are the same (e.g., truck
1 for the location 6), the two random numbers will be as the minimum one (0.01560744
in this case). Therefore, the truck will not leave for another location until it finishes the
current flood location. Therefore, the rank of 1 is repeated for the last two sub-genes in the
figure. Moreover, the ratio of truckload was converted from 0.51 to 1. The same is for the
second and the third locations. After that, truck 1 will serve location 1, which has the rank
of 2, with a random number of 0.21454631. Finally, location 4 then location 5 are served
by truck 1.
Sustainability 2023, 15, 9252
1 for the location 6), the two random numbers will be as the minimum one (0.01560744 in
this case). Therefore, the truck will not leave for another location until it finishes the current flood location. Therefore, the rank of 1 is repeated for the last two sub-genes in the
figure. Moreover, the ratio of truckload was converted from 0.51 to 1. The same is for the
second and the third locations. After that, truck 1 will serve location 1, which has the
rank
12 of 18
of 2, with a random number of 0.21454631. Finally, location 4 then location 5 are served
by truck 1.
Most probably, low random numbers (e.g., less than 0.20) and also large random
Most(e.g.,
probably,
low
random
numbers
(e.g., less than
0.20)
and also large random
numbers
greater
than
0.8) will
not be meaningful
for two
reasons:
numbers (e.g., greater than 0.8) will not be meaningful for two reasons:
1. In the optimal solution, the model tries to balance the load as much as possible on
In the optimal solution, the model tries to balance the load as much as possible on the
1.
the two trucks.
two trucks.
2. When converted into integers, the truckload for one vehicle might be zero or all the
2.
When converted into integers, the truckload for one vehicle might be zero or all the
truckload. For example, for location 4, if the percentage of truckload done by truck 3
truckload. For example, for location 4, if the percentage of truckload done by truck 3
is only 7%, the total truckload of location 3 is only 7, so 7×0.7 = 0.49, which can be
is only 7%, the total truckload of location 3 is only 7, so 7 × 0.7 = 0.49, which can be
rounded to zero. Therefore, the meaning of this is that only truck 1 serves location 4.
rounded to zero. Therefore, the meaning of this is that only truck 1 serves location 4.
Therefore, we generated random numbers between 0.2 and 0.8 to reduce the number
Therefore, we generated random numbers between 0.2 and 0.8 to reduce the number
of iterations to obtain the best solution. Table 5 shows the calculations of the end time for
of iterations to obtain the best solution. Table 5 shows the calculations of the end time for
truck 1 for the chromosome in Figure 5.
truck 1 for the chromosome in Figure 5.
Table5.5.Objective
Objectivefunction
functioncalculation
calculationfor
fortruck
truck11ininFigure
Figure5.5.
Table
Flood Location
6
6
11
44
55
Flood Location
Amj
8
3
2
4
Amj
t
8
3
2
4
1
1
0.5
0.7
t
1
1
0.5
0.7
p
8
5
4
3
p
8
5
4
3
(1 + t) Amj
16
16
66
33
6.8
6.8
(1 + t) Amj
End
Time (s+)
End Time (s+ )
16
16
22
22
25
25
31.8
31.8
worthmentioning
mentioninghere
herethat
thatsplitting
splittingthe
thejob
jobto
tobe
be performed
performedby
bytwo
two trucks
trucks affects
affects
ItItisisworth
the
objective
function
value
(total
weighted
flow
time).
Figure
6
shows
the
effect.
We
asthe objective function value (total weighted flow time). Figure 6 shows the effect. We
sumed that
durations in
in integer
integer numbers.
numbers. For
For
assumed
thattransportation
transportationtime
timeisiszero
zeroto
to have
have time
time durations
example,
location
1
is
dewatered
by
two
trucks.
A
completion
time
of
11
will
be
considexample, location 1 is dewatered by two trucks. A completion time of 11 will be considered
ered
the calculation
the objective
function.
is because
the penalty
be there
in
theincalculation
of theofobjective
function.
ThisThis
is because
the penalty
willwill
be there
as
as
long
as
there
is
flood
water
in
the
street.
This
is
why
we
cannot
calculate
the
objective
long as there is flood water in the street. This is why we cannot calculate the objective
function for
for different
different trucks
trucks independently.
independently. In
In location
location 5,
5, the
the two
two trucks
trucks (1
(1 and
and 2)
2) work
work
function
simultaneouslymost
mostof
ofthe
thetime.
time.
simultaneously
Figure
Figure6.6.Gantt
Ganttchart
chartfor
forthe
thesolution
solutionin
inFigure
Figure5,5,with
withzero
zerotransportation
transportationassumed.
assumed.
However, Figure 6 shows that different sequences can be for different trucks, but there
is a general sequence usually found in the optimal solution. For example, for truck 1 the
sequence is 6-1-4-5. Truck 2 has the sequence 2-5. For truck 3, the sequence is 1-3-4. That
means that a general sequence such as 2-6-1-3-4-5 can satisfy all three sequences. Most
probably, the optimal solution will have a general sequence that is appropriate for all the
trucks. Based on that, we can unify the sub-genes 4 and 5 and give them the same value.
This will reduce the search space and accelerate the GA. Based on the above discussion,
two tactics are used to make GA faster. The first one is to generate random numbers for the
truckload to be between 0.2 and 0.8 instead of generating random numbers between 0 and
1 the sequence is 6-1-4-5. Truck 2 has the sequence 2-5. For truck 3, the sequence is 1-3-4.
That means that a general sequence such as 2-6-1-3-4-5 can satisfy all three sequences.
Most probably, the optimal solution will have a general sequence that is appropriate for
all the trucks. Based on that, we can unify the sub-genes 4 and 5 and give them the same
value. This will reduce the search space and accelerate the GA. Based on the above
dis13 of
18
cussion, two tactics are used to make GA faster. The first one is to generate random numbers for the truckload to be between 0.2 and 0.8 instead of generating random numbers
between 0 and 1. The second tactic is to make sub-genes 4 and 5 have the same value so
1. The second tactic is to make sub-genes 4 and 5 have the same value so that one general
that one general sequence will control all the trucks in one chromosome.
sequence will control all the trucks in one chromosome.
The mating process is straightforward. Figure 7 shows a mating process example
The mating process is straightforward. Figure 7 shows a mating process example with
with a crossover point of 4.
a crossover point of 4.
Sustainability 2023, 15, 9252
Figure 7. Mating example for model 2.
Figure 7. Mating example for model 2.
For mutation, several random numbers are needed for each mutation process. The first
For mutation,
several random
numbers
needed
for each
mutation
The
one determines
the chromosome
to mutate.
A are
random
number
between
1 andprocess.
5 is needed
first
one
determines
the
chromosome
to
mutate.
A
random
number
between
1
and
5 is
to determine which sub-gene to mutate. For the first sub-gene, a new random number
needed
to
determine
which
sub-gene
to
mutate.
For
the
first
sub-gene,
a
new
random
between 0 and 1 is generated. If the number is below 0.5, a random vehicle number is
number between
0 andthen
1 is2generated.
If the number
is 1below
random vehicle
numassigned.
If it is larger,
random numbers
between
and T0.5,
are agenerated
to determine
ber
is
assigned.
If
it
is
larger,
then
2
random
numbers
between
1
and
T
are
generated
to
the location of the gene to exchange. Both the vehicle and its sequence are exchanged.
Sustainability 2023, 15, x FOR PEER REVIEW
14
of 19
determine
the location
of thesteps
gene using
to exchange.
Both the
vehicle and
sequence
Figure
8 shows
the mutation
these random
numbers.
The its
function
Unifare
(1,ex5)
changed.
Figure 8ashows
thenumber
mutation
steps using
random
numbers.
The function
means
generating
random
between
1 and these
5 using
a uniform
distribution.
Unif (1, 5) means generating a random number between 1 and 5 using a uniform distribution.
Select Chromosome
Select Sub-gene: Unif (1, 5)
Sub-gene 1
Sub-gene 2
Exchange
Sub-gene 3
No
Sub-gene 4
Sub-gene 5
Select gene to mutate: Unif (1, N)
trucks
Yes
Select two genes to exchange:
Unif (1, T) and Unif (1, T)
Sub-gene 1 or 2:
Sub-gene 3: Unif
Sub-gene 4 or 5:
Unif (1, T)
(0.2, 0.8)
Unif (0, 1)
Figure
8. 8.
Mutation
2.
Figure
Mutationsteps
stepsof
of Model
Model 2.
4. Results
andAnalysis
Analysis
4. Results
and
The summary results of the four GA models of the given data in Tables 1–3 can be
The summary results of the four GA models of the given data in Tables 1–3 can be
found in Table 6. The number of iterations differs from one run to another. The shown
found
The number
of iterations
differs
from
run
another.
The shown
onesin
areTable
those 6.
obtained
after running
the model
for the
firstone
time.
It istopossible
to obtain
ones
are those
obtained
afterFor
running
thefor
model
for1,the
first iftime.
It isitpossible
tocan
obtain a
a lower
or higher
number.
example,
model
maybe
we run
again, we
lower or higher number. For example, for model 1, maybe if we run it again, we can obtain
a lower number of iterations. In the first model, there are two trucks, and therefore, two
sequences are shown. Model 2 has 3 trucks, and therefore, three sequences are shown.
Having more trucks means faster work but at a higher cost. Model 2 has a lower objective
Sustainability 2023, 15, 9252
14 of 18
obtain a lower number of iterations. In the first model, there are two trucks, and therefore,
two sequences are shown. Model 2 has 3 trucks, and therefore, three sequences are shown.
Having more trucks means faster work but at a higher cost. Model 2 has a lower objective
function because there are three trucks and because job splitting can make work quicker.
However, job splitting adds coordination complexity and needs further planning.
Table 6. Models results.
Model Number
Objective Function
Number of Iterations Until the
Optimal Solution is Found
Optimal Sequence
1
2
1141.5
571.95
208
61
4-1-5-7 and 2-3-8-6
1-6-4-2-3, 6-2-5, and 1-4-3-5
It is obvious that the number of iterations is reasonable. In Model 2, the optimal
solution is not always guaranteed after a small number of iterations. However, a good
solution was always obtained. The detailed results of Model 2 are in Tables 7 and 8. We
assumed that the total number of available trucks is 3. Table 7 shows the results as can be
shown in the chromosomes in the cost function if all the rows are set beside each other.
The optimal solution indicates that each location is served by two trucks. Table 8 shows
the results understandably. It shows that truck 1 has 5 locations to serve. Truck 2 has 4
locations to serve. Truck 3 has four locations to serve. For example, truck 1 has the sequence
of 1-6-4-2-3. The numbers inside the parenthesis are the truckloads performed by the given
truck. For example, truck 1 has two truckloads to load and unload in the first location. The
sum of the truckloads must be the same as the estimated truckload of each location.
Table 7. Model 2 result.
Flood Location
First Truck
Second Truck
Truckload of the
First Truck
Sequence of the
First Truck
Sequence of the
Second Truck
Location 1
Location 2
Location 3
Location 4
Location 5
Location 6
1
1
1
1
2
1
3
2
3
3
3
2
0.4
0.4
0.4
0.29
0.8
0.38
1
4
5
3
3
2
1
2
3
2
4
1
Table 8. Model 2 results in a meaningful form.
Sequence * (Truckload)
Truck 1
Truck 2
Truck 3
Sum of truckload
1 (2)
1 (3)
5
6 (3)
6 (5)
8
4 (2)
4 (5)
7
2 (4)
2 (6)
10
3 (6)
3 (9)
15
5 (8)
5 (2)
10
* General sequence: 1-6-4-2-3-5.
Figure 9 shows the enhancements of the solution of model 2 over different iterations of
GA. The y-axis shows the objective function value, which measures the total weighted flow
time. It decreased from over 680 to 571.95. The solutions of model 2 converged quickly and
achieved a better result in fewer iterations.
Figure 10 shows the result of running Model 2 for 100 iterations when 30 flood locations
exist with 5 trucks. The input data of d, p, and t were set randomly. Great improvements
in objective function value were obtained during these iterations. The minimum value
was 5711.3. Different random sets were tried and even better enhancements were found.
Additionally, the solutions of Model 2 were robust and able to adapt to changing conditions.
Sum of truckload
5
8
7
10
15
10
* General sequence: 1-6-4-2-3-5.
Figure 9 shows the enhancements of the solution of model 2 over different iterations
of GA. The y-axis shows the objective function value, which measures the total weighted
of 18
flow time. It decreased from over 680 to 571.95. The solutions of model 2 15
converged
quickly and achieved a better result in fewer iterations.
Total weighted flow time
Sustainability 2023, 15, 9252
Sustainability 2023, 15, x FOR PEER REVIEW
16 of 19
Figure 9. Solution of model 2 for the given data.
Figure 10 shows the result of running Model 2 for 100 iterations when 30 flood locations exist with 5 trucks. The input data of d, p, and t were set randomly. Great improvements in objective function value were obtained during these iterations. The minimum
value was 5711.3. Different random sets were tried and even better enhancements were
found. Additionally, the solutions of Model 2 were robust and able to adapt to changing
Figure
9. Solution of model 2 for the given data.
conditions.
8000
Objective Function
7500
7000
6500
6000
5500
5000
0
20
40
60
Iterations
80
100
Figure10.
10.Result
Resultof
ofmodel
model22for
for30
30locations
locationsand
and55trucks.
trucks.
Figure
•
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the roads.
This study opens the door for future research in the same field, where the stochastic
nature of the problem can be investigated. New algorithms can be developed that
can better address the complexities of the problem.
The results of the study can be used to help decision-makers in similar contexts. If
the stochastic nature of the process is to be studied in the future, the dewatering process must be monitored and managed in order to keep the roads safe and efficient.
Sustainability 2023, 15, 9252
16 of 18
•
•
•
•
This study opens the door for future research in the same field, where the stochastic
nature of the problem can be investigated. New algorithms can be developed that can
better address the complexities of the problem.
The results of the study can be used to help decision-makers in similar contexts. If the
stochastic nature of the process is to be studied in the future, the dewatering process
must be monitored and managed in order to keep the roads safe and efficient. This can
be achieved through the use of sensors and other technologies to detect and correct
any potential issues. By doing so, the roads can remain safe and efficient for drivers.
Moreover, the idea of this study can be applied to other fields in which tankers are
used. For example, nondrinking water tankers are used extensively in the city for
many purposes.
The two models are new, and none of the previous studies considered the scheduling
problem of rainwater tankers. Therefore, there are no previous studies to compare
them with.
5. Conclusions
In this study, we investigated for the first time the water tanker scheduling problem.
The objective is to reduce the total weighted flow time. In other words, finishing the
dewatering process faster in more crowded streets. This is useful for safety and for smooth
movements on the street. We propose two MIP and GA models. The problem was divided
into two phases in the first model, and in the second model, a special type of job splitting is
proposed. Two tankers, at most, can work at the same location. New GA methods were
proposed to solve faster the MIP models. When GA is used, at least for small problems,
optimality is guaranteed. Results also showed that the number of iterations required to
find an optimal/good solution is reasonable. There are, however, some limitations to the
study. For example, the exactly needed truckloads are assumed to be known with certainty
based on experts’ judgment. The number of trucks is assumed as an input parameter.
Furthermore, the cost of more trucks was not taken into consideration. The number of
trucks can be considered as a decision variable in future work. Future research can also
look for faster algorithms for model 1. Another chance for future research is to merge
models 1 and 2. Therefore, consider two phases and two trucks, at most, in the same model.
Further, the penalty can be reduced whenever a truck finishes any amount of water on the
road. Future research can also investigate the stochastic nature of the process. For example,
during work, further rain can come and complicate the planning process.
Author Contributions: Conceptualization, M.A.; methodology, M.A.; software, M.A.; validation, B.S.
and N.G.; formal analysis, M.A.; investigation, M.A.; resources, B.S. and N.G.; data curation, M.A.;
writing—original draft preparation, M.A.; writing—review and editing, B.S. and N.G.; visualization,
B.S. and N.G.; supervision, B.S. and N.G.; project administration, M.A.; funding acquisition, B.S. All
authors have read and agreed to the published version of the manuscript.
Funding: This study received funding from King Saud University, Saudi Arabia, through researchers
supporting project number (RSP2023R145). Additionally, the APCs were funded by King Saud
University, Saudi Arabia, through researchers supporting project number (RSP2023R145).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments: Authors would like to thank King Saud University, Riyadh, Saudi Arabia, with
researchers supporting project number RSP2023R145.
Conflicts of Interest: The authors declare no conflict of interest.
Sustainability 2023, 15, 9252
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