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http://www.aimspress.com/journal/mbe
MBE, 20(8): 13754–13776.
DOI: 10.3934/mbe.2023612
Received: 06 April 2023
Revised: 16 May 2023
Accepted: 29 May 2023
Published: 16 June 2023
Research article
A new differential evolution using a bilevel optimization model for solving
generalized multi-point dynamic aggregation problems
Yu Shen1and Hecheng Li2,∗
1School of Computer Science and Technology, Qinghai Normal University, Xining 810008, Qinghai,
China
2School of Mathematics and Statistics, Qinghai Normal University, Xining 810008, Qinghai, China
*Correspondence: Email: lihecheng@qhnu.edu.cn.
Abstract: The multi-point dynamic aggregation problem (MPDAP) comes mainly from real-world
applications, which is characterized by dynamic task assignation and routing optimization with limited
resources. Due to the dynamic allocation of tasks, more than one optimization objective, limited
resources, and other factors involved, the computational complexity of both route programming and
resource allocation optimization is a growing problem. In this manuscript, a task scheduling problem
of fire-fighting robots is investigated and solved, and serves as a representative multi-point dynamic
aggregation problem. First, in terms of two optimized objectives, the cost and completion time, a new
bilevel programming model is presented, in which the task cost is taken as the leader’s objective. In
addition, in order to effectively solve the bilevel model, a differential evolution is developed based
on a new matrix coding scheme. Moreover, some percentage of high-quality solutions are applied
in mutation and selection operations, which helps to generate potentially better solutions and keep
them into the next generation of population. Finally, the experimental results show that the proposed
algorithm is feasible and effective in dealing with the multi-point dynamic aggregation problem.
Keywords: multi-point dynamic aggregation problem; multirobot system; task allocation; bilevel
optimization model; differential evolution
1. Introduction
The multi-point dynamic aggregation problem (MPDAP) [1] is widely used in resource allocation
and robot routing scheduling problems, such as post-disaster rescue [2], e-commerce logistics, forest
fire fighting [3], medical equipment scheduling [4], and other task allocation optimization problems [5,
6]. As a representative of the problems, robot routing optimization with dynamic task scheduling is
an interesting research issue in intelligent optimization community. For the purpose of improving the
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efficiency of completing tasks, the moving routes of the robots may often be adjusted according to the
observed tasks. Another characteristic of the problem is that there are multiple robots involved, that is
to say, this is a multi-robot task scheduling problem. Some interesting real-world applications, such as
e-commerce robots [5] and agricultural robots [6], have been presented in which the multi-robot task
allocation models are developed and solved. In these problems, one needs to consider that each may
change over time, the path of each robot, and the cooperation with other robots.
To better illustrate the problem, a two-robot and five-task case is present in Figure 1 and Figure 2.
Figure 1 is a schematic diagram of two robots cooperating to execute five tasks according to their own
driving path from the depot, in which the rectangle is the depot, the black circle represents the robot,
and the white circle stands for the task. The solid line is the driving path of robot 1, and the dashed
line is the driving path of robot 2. Figure 2 shows the time-demand change diagram of task 3 which
needs to be executed by the robots. In Figure 2, point Agives the initial state value of task 3, point B
provides the total task demand facing robot 1 and robot 2 when they reach task point 3, and the task
demand at point Capproaches 0, indicating that task 3 has been cooperatively completed by both robot
1 and robot 2. It can also be seen from Figure 2 that the cumulative time from the depot to completing
task 3 is 5.3. One methodology to simplify the problem is to minimize the time cost at which all five
tasks are completed by the two robots.
task1
task2
depot
task3
task4
task5
robot1
robot2
Figure 1. Task allocation problem.
0123456
time
0
1
2
3
4
5
6
7
8
9
demand
A(0,1.09)
B(4,8.05)
C(5.3,0)
Figure 2. Time-demand curve for task3
in the figure 1.
The above-mentioned multi-point dynamic aggregation problem is a combination of path optimiza-
tion problems and the task assignment of multi-robot systems. It not only needs to plan the path of each
robot to execute the task, but also reasonably adjust the cooperation between robots to complete the
tasks at which the time-demanding amount may be dynamically changing. Therefore, the multi-point
dynamic aggregation problem has the following challenges. First, in order to complete a task requires
the cooperation of robots, each robot needs to consider its own path as well as plans of other robots
during the task execution. Second, the time each robot requires to complete a certain task depends
on the growth rate of task demand at each task, which is a dynamically decision-making procedure.
Third, the optimization of both the traveling routes of robots and other objectives has to be made in a
multi-factor interactive environment involving dynamic task allocation and cooperation among multi-
ple individuals, etc.
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Despite the computational complexity of the problem, the wide range of practical applications has
stimulated the study of modeling and solution methods of the problem. In [7], a model of forest fire
suppression was presented as a multiple permutation combinatorial optimization problem, named MP-
DAP, and a distribution estimation algorithm using an effective encoding method was proposed for
this problem. The given experimental show that the proposed algorithm can find a better solution for
MPDAP than other compared approaches. In [8], a nonlinear agent routing problem in multi-point
dynamic task (ARP-MPDT) was developed, which was based on a node histogram model (NHM), an
edge histogram model (EHM) and probabilistic modeling distribution estimation algorithm (EDA), in
which the ratio of NHM and EHM probability models can be adjusted adaptively. Gao et al. [9] rep-
resented each robot task by multiple row access sequence coding which designed a kind of heuristic
rule of implicit decoding strategy to simplify the problem from the mixed-discrete variable optimiza-
tion, and was based on a one-step and multi-step local search and presented a memetic algorithm for
solving the problem. Hao et al. [10] combined the advantages of differential evolution algorithms and
distribution estimation algorithms, and then proposed a hybrid algorithm. The proposed algorithm was
executed on MPDAP examples of different sizes, and the simulation results show that compared with
single differential evolution or distribution estimation algorithms, the hybrid method can solve this
problem more efficiently. In an ant colony framework, in order to improve the efficiency of the solu-
tion construction, a new pheromone matrix and pheromone updating mechanism were proposed [11].
In order to reduce the search space and delete some bad solutions, Gao et al. [11] designed a heuristic
value evaluation function in solution construction, and a pheromone repair based mechanism was in-
vestigated to enhance the ability to build viable solutions, and a local search was adopted to improve
the balance between exploration and exploitation.
It should be noted that most of existing models for MPDAPs work to minimize the completion time
of all tasks, in which the number of robots is predetermined. A recurring question is ”how may robots
are best?”, that is to say, from the point of view of controlling total costs involving robots, time costs
and material damage, how does a decision-maker optimize the number of dispatching robots and the
traveling lines of them? Different from the above literatures, in the manuscript, a new bilevel program-
ming problem is provided and solved by designing a new encoding scheme and efficient evolutionary
operators. Some main innovations of this manuscript are presented as follows:
1) In the proposed model, the total cost of performing all tasks, instead of only time cost, is taken
into account, in which the number of robots need to be optimized. Hence, the existing model with a
predetermined number of robots is simply a special case of the proposed model. In the proposed model,
both the robot cost and completion time are located in different levels and hierarchically optimized.
2) A new matrix encoding scheme is developed for all follower’s solutions, which is simpler in form
and more efficient than most of the existing encoding methods.
3) Both novel two-strategy mutation and selection schemes are designed and adopted in different
evolution stages, which helps to find better potential solutions.
The rest of the manuscript is arranged as follows. Section 2 introduces bilevel models and differ-
ential evolution methods. Section 3 introduces the MPDAP model. The detailed algorithm design is
presented in Section 4. The experimental studies are implemented in Section 5. Section 6 provides a
discussion. Finally, Section 7 concludes this manuscript.
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2. Preliminaries
2.1. Bilevel optimization models
Bilevel programming is a hierarchical optimization problem, in which variables are divided and
located at two levels, the leader’s and follower’s levels, respectively. A general bilevel optimization
model can be formulated as
min
x∈XF(x,y)
s.t G(x,y)≤0
min
y∈Yf(x,y)
s.tg(x,y)≤0
(2.1)
where
min
x∈XF(x,y)
s.t G(x,y)≤0(2.2)
and
min
x∈Yf(x,y)
s.tg(x,y)≤0(2.3)
are called the leader’s and follower’s problems, respectively. F(x,y) and f(x,y) are the leader’s and
follower’s objective functions, respectively, G(x,y) and g(x,y) are the constraints of the leader’s and
follower’s problems, respectively, and , xand yare called the leader’s and follower’s variables, respec-
tively. The bilevel programming problem is widely used in many practical fields, such as economics,
engineering science, operation research, and optimal control [12], and has a decision-making system
with a hierarchical (nested) optimization process, in which multiple decision-makers with different pri-
orities locates at different decision-making levels. Generally speaking, the priority of decision-makers
at the higher level is higher than that of decision-makers at the lower level. The decision-makers at the
higher level regulate the decision-making process at the lower level through optimization procedures,
and the decision of the follower can also affect the optimization result of the leader’s problem.
The computational difficulty of solving the bilevel optimization problem lies in the large amount
of computation caused by frequently solving the follower’s problems. At present, the methods to
solve bilevel optimization problems mainly include the traditional optimization method using gradient
descent, swarm intelligent approaches, and hybrid technologies. The traditional optimization meth-
ods include the single level reduction method, the branch and bound method, the penalty function
method, the trust region method, and so on. Sinha et al. [13] used Karush-Kuhn-Tucher (K-K-T) con-
ditions to transform the bilevel optimization model into a single-level optimization problem, though
the transformed model always causes a complex constraint restriction. For integer bilevel optimization
problems, Liu et al. [14] proposed a branch and bound method based on a relax and check procedure.
Li et al. [15] proposed a numerical method for solving bilevel optimization problems with non-convex
follower’s problems, in which, the follower’s problem is replaced by an equivalent substitution through
K-K-T conditions. Joseph et al. [16] proposed a bilevel optimization model for classification feature
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selection problems and designed a genetic algorithm based on derivatively-free optimization. Li et
al. [17] studied a class of linear fraction bilevel optimization problems where the coefficients and the
right vector of the objective function are interval numbers and proposed an evolutionary algorithm
based on optimality conditions. Aboelnage et al. [18] proposed an improved algorithm based on a
new selection method and a chaotic search method for bilevel optimization problems. Goshu et al. [19]
proposed a meta-heuristic algorithm based on a random space system sampling technique to solve the
stochastic bilevel optimization problem. The hybrid optimization method combines the advantages
of various algorithms to deal with the bilevel optimization problem, and shows a good performance.
Islam et al. [20] proposed a memetic algorithm combining global search and local search. Yousria
et al. [21] proposed a hybrid algorithm combining an improved genetic algorithm and chaotic search
techniques. Sinha et al. [22] proposed a bilevel optimization algorithm with a multi-valued iterative
approximation, which uses the follower’s reaction set mapping to reduce the computation. Molai [23]
used an extension principle for fuzzy multi-objective linear bi-level optimization problems.
Most of the existing methods are efficient in solving bilevel programming problems with small scale
convex and differentiable functions. However, when the scale of the problem is sharply increased and
non-convex, either non-differential functions or discrete variables are involved and the performance of
these approaches always degenerates rapidly and can’t be used to deal with the problem of this kind.
2.2. Differential evolution
Differential evolution (DE) is a very popular evolutionary algorithm paradigm proposed by Storn
and Price [24]. It contains four processes: initialization, mutation, crossover, and selection.
Initialization: There are Nindividuals generated randomly in a feasible region.
P(T)={X1,X2,· · · ,Xm,· · · ,XN}(2.4)
where Xmis the mth individual.
Mutation: The mutation operation is the main way for populations to generate new individuals,
which can increase population diversity. For each individual Xm, a mutation vector Vmis generated.
Some common mutation operators are as follows
DE/rand/1/bin:
Vm=Xr1+Fs f (Xr2−Xr3) (2.5)
DE/best/1/bin:
Vm=Xbest +Fs f (Xr1−Xr2) (2.6)
DE/rand/2/bin:
Vm=Xr1+Fs f1(Xr2−Xr3)+Fs f2(Xr4−Xr5) (2.7)
DE/best/2/bin:
Vm=Xbest +Fs f1(Xr2−Xr3)+Fs f2(Xr4−Xr5) (2.8)
DE/rand-to-best/1/bin:
Vm=Xr1+rand(0,1)(Xbest −Xr1)+Fs f2(Xr2−Xr3) (2.9)
where r1,r2,r3,r4, and r5are five different integers chosen randomly from [1,N], Xbest is the best
individual in the population, rand(0,1) is a uniformly distributed random number from [0,1], and Fs f ,
Fs f1, and Fs f2are scaling factors.
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Crossover: Using binomial crossover to generate crossover offspring Um.
Selection: The better one between Xmand Umis selected into the next generation.
DE uses the differential information of multiple individuals in a population to achieve perturbation
of evolved individuals. As s one of the effective swarm intelligent algorithms, DE has been success-
fully applied to many practical problems [25]. Wang et al. [26] used a new encoding method based
differential evolution to solve the distribution problem of wind farms. Aiming at power optimization
problem of power system, Chi et al. [27] adopted a differential evolution algorithm based on different
evolutionary operations. Additionally, other interesting applications can be found, such as economic
or emission dispatch problem [28], image segmentation [29], and task scheduling in cloud computing
environment [30]. As a bilevel programming model with discrete variables, the proposed MPDAP in
the manuscript is very hard to solve. In view of the good performance of differential evolution in deal-
ing with complex optimization problems, this paper developed an improved version of this algorithm
to solve the proposed MPDAP model.
3. Multi-point dynamic aggregation problem
3.1. Demand state model for task in MPDAP
From the introduction, it can be seen that in a MPDAP, the task demand of each task (e.g. fire point)
may dynamically increase over time. The relationship between task demand and time is suggested by
the following equations in [11]:
qt
i=q0
i+αit(3.1)
where, qt
irepresents the demand at task iat time t, the initial demand of task iis q0
i, and αiis the
inherent increment rate of task i.
3.2. The model of the MPDAP
The workload of the robot rto solve the task ican be expressed by the following formula:
ttaski=Staski
vr
task
(3.2)
where, Staskirepresents the total amount of work that robot does, vr
task is the work efficiency, and ttaski
stands for the time.
Spath ji is the path length from task jto task i, the speed of the robot ris vr
path, and the time taken
from task jto task iis tpath ji . The formula is as follows:
tpath ji =Spath ji
vr
path
(3.3)
In [11], the objective function in the MPDAP model is
min f=max
i=1,2,··· ,M1
ti(3.4)
where, tiis the completion time of task i. In fact, in a real-world planing problem, it is very important
to control overall cost, not just time cost. In this manuscript, a more general case is taken into account,
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which involves the cost of operation time, damage caused by fire, and robot cost. In the model, once
the number of robots are determined, one can program the route and assign a task for each robot. In
this procedure, a hierarchical optimization process arises. As a result, a bilevel programming model is
developed as follows:
min F=F1+F2+F3
min f=max
i=1,2,··· ,M1
ti
(3.5)
where, the leader’s objective function Fstands for the cost of all the robots to complete the tasks and
the amount of damage in the tasks; the follower’s objection function fis the maximum completion
time of all the tasks.
F1=
M1
X
i
λ1qti
i(3.6)
F2=
M2
X
r
c0
r(3.7)
F3=
M2
X
r
cr(min max
i=1,2,··· ,M1
ti) (3.8)
ti=tj+tpath ji +ttaski(3.9)
Equation (3.6) represents the total amount of damages at all M1tasks when all tasks are completed,
in which the total amount of damage at task iis λ1times the task demand at task i. Equation (3.7) is the
state-determined cost of renting (or buying) M2robots used in the tasks, here, c0
ris the inherent cost of
robot r. Equation (3.8) is the running cost of all the robots and cris the unit running cost of robot r.
As is known, model (3.4) is simply the follower’s problem of model (3.5). The number of robots
and inherent costs in model (3.5) are fixed, which is degenerated to model (3.4). Therefore, model
(3.5) is more general and extensive than model (3.4).
In order to make the model of multiple robots and tasks more explicit, relevant assumptions in [11]
are also adopted in this manuscript:
1) There are M2robots, and their initial location is the depot. The cost of robots of the same size is
the same, and they have the same traveling speed vpath and the work efficiency vtask.
2) There are M1tasks in different positions, the tasks at different locations may have the same or
different task demand.
3) In the working environment, there are no obstacles, and each robot does not have to consider the
collision with others. Once the driving path is determined, the robot can drive.
4) The starting time that the robots leaves the depot is denoted as 0.
5) For each task, the number of ways into the task point equals to that of ways out.
6) Each task is performed by at least one robot.
7) Each task is executed by a robot at most once.
8) Once the current task is completed, the robot can move on to the next.
9) The moving path of each robot can be planned in advance before performing tasks.
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4. Algorithmic design
In the proposed bilevel model (3.5), the leader’s problem is to optimize the number of robots as-
signed to tasks. Once the number of the robots is determined, the follower needs to optimize the
moving routes of robots. Since the optimization procedure is hierarchical and most of the compu-
tational amounts mainly come from the solutions of the follower’s problem, the manuscript is first
focused on the follower’s algorithm design. Finally, by embedding the follower’s algorithm, a novel
bilevel differential evolution (NDE) is presented for the bilevel model (3.5).
4.1. Algorithmic design of the follower’s problem
This section mainly introduces the main schemes, as well as the follower’s algorithm. First of all, a
new encoding scheme is developed, which can more efficiently express the robot’s traveling path and
completion tasks than others. In addition, a two-strategy mutation is provided by using the mean of
locations of the top individuals, which can efficiently avoid the directional mislead from a single good
point.
4.1.1. Encoding /Decoding strategies
In an MPDAP tasks planning problem with M1tasks and M2robots, a feasible solution is encoded
as an M2×(M1+1) matrix X:
Xm=
1xm
11 xm
12 · · · xm
1M1
1xm
21 xm
22 · · · xm
2M1
.
.
..
.
..
.
..
.
..
.
.
1· · · xm
ri · · · xm
rM1
.
.
..
.
..
.
..
.
..
.
.
1xm
M21xm
M22· · · xm
M2M1
(4.1)
where the elements in the first column of the matrix (4.1) are all 1, indicating that every robot starts
from the depot. From the second column to the last column, the elements are 0 or 1, as shown in (4.2).
(4.3) indicates that, for each task, the total ability of the robots executing it must be greater than its
inherent increment rate. Otherwise, the task can never be completed.
xri =
1,i f robot r goes to task i and completes the task i
0,otherwise (4.2)
M2
X
r=1
xrivr
task > αi(4.3)
The matrix encoding strategy is presented in a simpler form, where the elements of the matrix are
either 0 or 1, that is to say, there are only two states, go or not. In the encoding scheme, the driving
path of each robot is represented by a row, and a feasible solution is equivalent to a scheduling plan
that can efficiently complete all tasks. Furthermore, a binary encoding structure can be conveniently
used in evolutionary operations and can efficiently search for potential path combinations.
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The decoding procedure is executed as follows: in each row of (4.1), value 1 means the robot visits
the task, whereas value 0 represents the robot never visits the task. The order of visiting tasks is
determined by
VOtaski=αi
spathi
(4.4)
where αiis the inherent increment rate of task iand spathiis the path length from depot to task i. A task
with a larger αiand a smaller spathishould be visited preferentially, Hence, VOtaskican be taken as an
index by which the tasks can be chosen by a probability choice scheme, one by one.
For example, the example in Figure 1 would be encoded as follows:
"1 1 0 1 1 0
1 0 1 1 0 1#(4.5)
where the first row [1 1 0 1 1 0] means that tasks [depot,task1,task3,task4] needs to be visited sequen-
tially. The values of the relevant parameters of the tasks in Table 1. According to the order V Otask3>
VOtask1>V Ota sk4, the visiting path of robot1is determined as depot →task3→task1→task4.
Table 1. The values of the relevant parameters of the tasks in Figure 1.
Parameters task1ta sk2task3task4ta sk5
α0.3 0.3 0.5 0.2 0.2
S3 3 4 8 8
VOt ask 0.1 0.1 0.125 0.025 0.025
4.1.2. Evolutionary operators
I. Mutation operator
In the proposed follower’s algorithm, the total two mutation strategies are adopted in early and later
evolution stages, respectively. The first mutation operator is designed as follows:
Mutation operator 1, DE/rand −to −meanbetter/1/bin:
Vm=Xr1+rand(0,1)(Xmeanbetter −Xr1)+Fs f (Xr2−Xr3) (4.6)
In the mutation operator, a group of top individuals are chosen to provide an evolutionary direction
for individuals. Different from the mutation operator 2 (as below), the mutation operator 1 can keep the
diversity of offspring since the heuristic information comes from a group of better individuals, instead
of a single point. The advantage of the scheme is that more than one better point can provide a more
stable evolutionary direction than that a single point does.
In the later stage of evolution, the population trends to convergence and then only a single best
individual can work well. As a result, the following mutation operation 2 is directly adopted.
Mutation operation 2, DE/rand −to −best/1/bin:
Vm=Xr1+rand(0,1)(Xbest −Xr1)+Fs f (Xr2−Xr3) (4.7)
where Xr1,Xr2, and Xr3are randomly selected individuals in the population, rand(0,1) represents a
random number that follows an uniform distribution in [0,1], Fsf ∈[0,2] is the scaling factor, Xmeanbest
is the average position of some top individuals, and Xbest is the best individual found so far.
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The offspring of the mutation operator can be described as
Vm=
1vm
11 vm
12 · · · vm
1M1
1vm
21 vm
22 · · · vm
2M1
.
.
..
.
..
.
..
.
..
.
.
1· · · vm
ri · · · vm
rM1
.
.
..
.
..
.
..
.
..
.
.
1vm
M21vm
M22· · · vm
M2M1
(4.8)
II. Crossover operator
Through the binomial crossover, a trial vector Umis generated based on Xmand Vm.
Um=
1um
11 um
12 · · · um
1M1
1um
21 um
22 · · · um
2M1
.
.
..
.
..
.
..
.
..
.
.
1· · · um
ri · · · um
rM1
.
.
..
.
..
.
..
.
..
.
.
1um
M21um
M22· · · um
M2M1
(4.9)
um
ri =
vm
ri ,i f rand j≤CR or j =jrand
xm
ri ,otherwise (4.10)
where m=1,2,· · · ,NP,CR ∈[0,1] is the crossover control parameter, and randjis a random number
[0,1].
III. Individual repair operator
During the evolution of the individual, there may be cases where none of the robots can complete
their allocated tasks, leading to an infeasible solution. For example, given an MPDAP instance with
two robots and five tasks, the inherent increment rate of a task is larger than the individual ability of
both robots. Thus, the robots have to go to the same task and complete it. If only a single robot is
asked to execute a task with a large inherent increment rate separately, then such a task will not be
completed. To address this issue, a individual repair operator is developed to reallocate the robots to
make the solution feasible. The mutation and crossover operators only affect the second to last column
of the individual matrix Xmand Vm, so the individual repair operator will only target these columns as
well.
In the offspring individual matrix Um, the elements in the first column are all 1, but the elements in
the other positions may not take values of 0 or 1, and perhaps do not satisfy the constraint (4.3). The
repair process is as follows:
Step 1. The elements in the individual matrix are made feasible, i.e., the elements greater than 1
become 1, the elements less than 0 become 0, and others are rounded off.
Step 2. Verify whether the elements of each column satisfy constraints (4.3), if so, turn to step 4,
otherwise, turn to step 3.
Step 3. Randomly select the element in each column that is 0 and change it to 1 until the element in
that column satisfies constraint (4.3).
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Step 4. End the repair process.
IV. Selection operator
In the process of evolution, two selection operators are used. One is the same as in the original
differential evolution, which is denoted by Selection operator 1 and expressed as follows:
Xi=
Ui,i f f f it(Ui)≤ff it (Xi)
Xi,otherwise (4.11)
where, ff it, as the fitness, is the follower’s objective function .
The other is newly provided as a selection operator 2, which can be finished as follows: in all
individuals, parents and offspring, some top individuals are first selected into the next generation of
population, and then the rest are chosen by the procedure of selection operator 1. It follows that
the selection operator 2 focuses on better individuals instead of diversity kept by parents. Hence the
selection operator 2 is helpful to speed up the convergence of the algorithm.
In early evolution, the selection operator 1 is executed for the purpose of maintaining diversity. In
order to improve the exploitation capability of the algorithm, the selection operator 2 is utilized in the
later stage.
4.1.3. Pseudo-code and flowchart of the follower’s algorithm
Based on the mentioned-above design, the pseudo-code of the follower’s algorithm is presented in
Algorithm 1.
Algorithm 1 Procedure of the proposed follower’s algorithm
Require: Tf
max : maximum running times; benchmark instance; Nf:population size; Y:leader’s variable (act as parameters for follower’s
problem)
Ensure: output the set Ato get the optimal solution X∗
1: generate initial population Pf(Tf)={X1,X2,· · · ,XNf};Tf←0.
2: store the optimal individual to the set Aaccording to the fitness value of the individuals in Pf(Tf).
3: while Tf≤αTf
max do
4: the offspring population Of(Tf) is obtained by the mutation operator 1 and crossover operator of population Pf(Tf).
5: the next generation population Pf(Tf+1) is obtained from population Pf(Tf) and offspring population Of(Tf) according to
individual repair operator and the selection operator 1.
6: update the set Aaccording to the fitness value of the individuals in Of(Tf).
7: Tf=Tf+1.
8: end while
9: while αTf
max <Tf≤Tf
max do
10: the offspring population Of(Tf) is obtained by the mutation operator 2 and crossover operator of population Pf(Tf).
11: the next generation population Pf(Tf+1) is obtained from population Pf(Tf) and offspring population Of(Tf) according to
individual repair operator and the selection operator 2.
12: update the set Aaccording to the fitness value of the individuals in Of(Tf).
13: Tf=Tf+1.
14: end while
4.2. A novel differential evolution for the bilevel model (NDE)
In the bilevel optimization algorithm for solving the MPDAP, the leader’s objective function is taken
as the fitness function, and a classical differential evolution is adopted to optimize the variables of the
leader’s problem. The pseudo-code of the bilevel optimization algorithm is presented in Algorithm 2.
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Algorithm 2 NDE for solving the bilevel optimization model
Require: Nl: population size; Tl
max : maximum running times;
Ensure: the best solution Y∗best
1: generate initial population Pl(Tl)={Y1,Y2,· · · ,YNu};Tl←0.
2: use Algorithm 1 to solve the corresponding follower’s problem for each individual from the population Pl(Tl).
3: store the optimal individual to the set Baccording to the fitness value of the individuals in Pl(Tl).
4: while Tl≤Tl
max do
5: the offspring population Ol(Tl) is obtained from Pl(Tl) by the mutation operator and the crossover operator in classical DE.
6: the next generation population Pl(Tl+1) is obtained from population Pl(Tl) and offspring population Ol(Tl) according to the
selection operator in classical DE.
7: use Algorithm 1 to solve the corresponding follower’s problem for each individual in population Pl(Tl+1).
8: update the set Baccording to the fitness value of the individuals in Pl(Tl+1).
9: Tl=Tl+1.
10: end while
5. Simulation
5.1. Comparison with other methods
All 50 benchmark instances are taken from [11]. In these 50 benchmark instances, the number of
robots is fixed, and the task demand function is a linear function. The differences between groups of
benchmark instances are shown in Table 2. M1is the number of tasks and M2is the number of robots.
In order to make the experimental results well presented, the benchmark instances were divided into
3 groups, namely Group 1, Group 2, and Group 3. All the instances are called by their group name,
number of robots, number of tasks, and the ratio between the sum of all the task inherent rates and the
sum of all the robot abilities. For example, the benchmark instance 1 is named as G15 4 0.39, where
G1denotes the group of the instance, 5 is the number of robots, 4 is the number of tasks, and 0.39 is
the ratio. In order to make the comparison fair and verify the effectiveness of the NDE, the parameter
settings are taken similar to the benchmark instances, and the stopping criterion is set is the same as
the competitors. The proposed algorithm is run independently for 30 times, and the computational
results are recorded in Tables 3-8. The data of MA [9] based on two variants are shown in columns
MA-MLS and MA-OLS in Tables 3, 5, and 7. The data of EDA [10] are shown in column EDA in
Tables 3, 5, and 7. The data of ILS [32] are presented in column ILS in Tables 4, 6, and 8. The data
of AC-ACO [11] are provided in column AC-ACO in Table 4, 6, and 8. Column NDE provides the
computational results by NDE.
Table 2. Differences between groups of benchmark instances.
Instance The number of instances Range of M1Range of M2Range of M1+M2
Group 1 27 [4,40] [3,30] [0,50)
Group 2 6 [10,60] [15,40] [50,95)
Group 3 17 [15,120] [20,120] [95,180]
In Tables 3-8, the comparison data include the means and standard deviations of the completion
time of tasks, and ∗means the corresponding method cannot obtain a feasible solution in a limited
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period of time. The numbers in parentheses indicate the rank-order of all compared approaches on a
benchmark instance. The smaller the rank value, the better the algorithm. In addition, some visual
comparisons are also provided by Figure 3 (the Group 1), Figure 4 (the Group 2), and Figure 5 (the
Group 3). For the case that some algorithms fail to find the optimal solution on some instances, a
maximum value is used on the figure to ensure the visuality of figures.
The results of the Group 1 are shown in Table 3 and Table 4. NDE is slightly worse than MA-MLS in
G115 20 0.67, G117 23 1.71, G120 20 0.58, and G120 20 0.967. However, NDE is better than the
comparison algorithm in the remaining 23 instances in Table 3. In Table 4, NDE provides worse results
for 11 instances, but better results for the remaining 16 instances. Among the 27 benchmark instances
in Group 1, the rank-order values of NDE are 31 and 41, respectively. In Table 4, the rank-order values
of both AC-ACO and NDE are 41, indicating that they both have similar algorithm performances.
Compared with other algorithms except AC-ACO, the rank values of NDE are the smallest.
Table 3. Comparison (I) of experimental results on the Group 1.
Instance MA-MLS MA-OLS EDA NDE
Mean Std Mean Std Mean Std Mean Std
G15 4 0.39 1.07E+2(4) 1.9E+0 1.06E+2(3) 1.5E+0 1.05E+2(2) 1.2E-1 1.03E+2(1) 1.3E+0
G15 5 1.66 1.22E+3(3) 4.6E+1 1.17E+3(2) 3.2E+1 1.31E+3(4) 5.9E+1 1.16E+3(1) 3.0E+1
G13 10 1.51 9.52E+2(3) 2.1E+1 9.22E+2(2) 3.6E+1 1.03E+3(4) 2.9E+1 8.76E+2(1) 2.5E+0
G13 15 5.03 1.09E+5(3) 4.4E+4 5.98E+4(2) 1.3E+4∗(4) ∗2.75E+4(1) 7.4E+2
G15 10 0.93 4.33E+2(2) 1.1E+1 4.20E+2(1) 1.2E+1 4.78E+2(3) 9.4E+0 4.09E+2(1) 7.5E+0
G110 5 1.39 6.00E+2(3) 2.2E+1 5.87E+2(2) 2.6E+1 6.53E+2(4) 2.7E+1 5.65E+2(1) 1.1E+1
G15 10 3.67 3.12E+4(3) 6.3E+3 2.23E+4(2) 2.9E+3∗(4) ∗1.01E+4(1) 2.8E+3
G15 20 4.36 5.24E+4(3) 7.2E+3 3.74E+4(2) 4.3E+3∗(4) ∗2.03E+3(1) 4.5E+2
G110 10 3.79 3.53E+4(3) 7.8E+3 3.19E+4(2) 6.8E+3∗(4) ∗7.79E+3(1) 3.2E+2
G111 11 1.28 3.05E+2(2) 1.7E+1 3.10E+2(3) 1.0E+1 4.06E+2(4) 1.7E+1 2.50E+2(1) 3.0E+1
G130 5 0.46 1.50E+2(1) 9.7E-1 1.50E+2(1) 6.9E-1 1.55E+2(2) 1.7E+0 1.50E+2(1) 1.7E+0
G115 10 1.17 3.39E+2(2) 1.0E+1 3.59E+2(3) 9.0E+0 4.19E+2(4) 1.1E+1 3.25E+2(1) 1.1E+0
G110 15 1.3 6.28E+2(2) 9.7E+0 6.44E+2(3) 1.0E+1 8.07E+2(4) 3.3E+1 6.15E+2(1) 4.7E+0
G120 10 0.47 1.04E+2(3) 1.9E+0 1.03E+2(2) 1.7E+0 1.13E+2(4) 2.5E+0 9.77E+1(1) 1.5E+0
G120 10 0.96 3.41E+2(2) 7.0E+0 3.61E+2(3) 4.5E+0 4.05E+2(4) 1.2E+1 3.31E+2(1) 5.0E+0
G120 10 0.94 2.73E+2(2) 8.5E+0 2.79E+2(3) 3.7E+0 3.12E+2(4) 6.5E+0 2.38E+2(1) 3.2E+0
G15 40 3.95 1.51E+4(3) 7.3E+2 1.43E+4(2) 5.1E+2 2.73E+4(4) 2.2E+3 1.23E+4(1) 5.6E+2
G110 20 6.04 ∗(2) ∗ ∗(2) ∗ ∗(2) ∗8.94E+4(1) 4.23E+3
G130 10 0.65 1.83E+2(2) 3.9E+0 1.84E+2(3) 3.4E+0 1.97E+2(4) 4.6E+0 1.68E+2(1) 2.2E+0
G115 20 0.67 3.51E+2(1) 7.0E+0 3.67E+2(3) 1.8E+0 3.92E+2(4) 6.1E+0 3.60E+2(2) 4.1E+0
G130 10 1.34 4.14E+2(2) 1.6E+1 4.58E+2(3) 8.0E+0 5.35E+2(4) 2.9E+1 4.02E+2(1) 2.3E+0
G115 20 5.98 ∗(2) ∗ ∗(2) ∗ ∗(2) ∗7.02E+4(1) 1.9E+3
G117 23 1.71 4.07E+2(1) 2.4E+1 6.10E+2(3) 2.5E+1 8.12E+2(4) 4.1E+1 4.58E+2(2) 4E+0
G120 20 0.58 2.74E+2(1) 5.7E+0 2.88E+2(3) 2.5E+0 3.05E+2(4) 4.4E+0 2.86E+2(2) 2.3E+0
G120 20 0.97 1.92E+2(2) 1.1E+1 2.53E+2(3) 6.3E+0 2.82E+2(4) 1.2E+1 1.45E+29(1) 1.3E+0
G120 20 0.967 4.01E+2(1) 7.8E+0 4.26E+2(3) 5.7E+0 4.80E+2(4) 1.3E+1 4.08E+2(2) 4.5E+0
G115 30 2.16 1.68E+3(2) 7.4E+1 2.09E+3(3) 5.8E+1 2.44E+3(4) 1.0E+2 1.41E+3(1) 2.1E+1
Rank value 60 - 66 - 99 - 31 -
The computational results on the Group 2 are shown in Table 5 and Table 6. In Table 5-6, NDE
has slightly worse results for G240 15 0.67, but better results in the remaining 5 instances. For 6
benchmark instances in the Group 2, the rank-order of the NDE are 9 and 8 , respectively. Compared
with other algorithms, NDE has a better performance in Group 2.
The computational results on Group 3 are shown in Table 7 and Table 8. In Table 7, the results of
NDE are all better than those of the comparison algorithm. Although Table 8 displays that NDE is
slightly worse than AC-ACO in G360 30 1.14 and G380 80 0.54, NDE is better than the comparison
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Table 4. Comparison (II) of experimental results on the Group 1.
Instance ILS AC-ACO NDE
Mean Std Mean Std Mean Std
G15 4 0.39 1.06E+2(2) 1.3E+0 1.06E+2(2) 1.3E+0 1.03E+2(1) 1.3E+0
G15 5 1.66 1.20E+3(2) 5.7E+1 1.23E+3(3) 2.9E+1 1.16E+3(1) 3.0E+1
G13 10 1.51 8.99E+2(3) 3.2E+1 8.77E+2(1) 1.5E+1 8.76E+2(1) 5E+0
G13 15 5.03 4.00E+4(3) 7.4E+3 2.73E+4(1) 10.0E+1 2.75E+4(2) 7.4E+2
G15 10 0.93 4.15E+2(2) 9.4E+0 4.09E+2(1) 8.2E+0 4.09E+2(1) 7.5E+0
G110 5 1.39 6.01E+2(3) 3.6E+1 5.68E+2(2) 1.7E+1 5.65E+2(1) 1.1E+1
G15 10 3.67 1.83E+4(3) 1.8E+3 1.31E+4(2) 7.1E+2 1.01E+4(1) 2.8E+3
G15 20 4.36 3.35E+4(3) 4.7E+4 1.79E+4(2) 1.0E+3 2.03E+3(1) 4.5E+2
G110 10 3.79 1.76E+4(3) 2.5E+3 7.44E+3(1) 3.7E+2 7.79E+3(2) 3.2E+2
G111 11 1.28 3.36E+2(3) 2.3E+1 2.79E+2(2) 1.1E+1 2.50E+2(1) 3.0E+1
G130 5 0.46 1.50E+2(1) 1.8E+0 1.50E+2(1) 1.9E+0 1.50E+2(1) 1.7E+0
G115 10 1.17 3.61E+2(3) 2.1E+1 3.42E+2(2) 8.0E+0 3.25E+2(1) 1.1E+0
G110 15 1.3 6.55E+2(3) 3.7E+1 6.30E+2(2) 1.3E+1 6.15E+2(1) 4.7E+0
G120 10 0.47 9.85E+1(3) 1.8E+0 9.67E+1(1) 1.7E+0 9.77E+1(2) 1.5E+0
G120 10 0.96 3.61E+2(3) 2.4E+1 3.41E+2(2) 7.2E+0 3.31E+2(1) 5.0E+0
G120 10 0.94 2.70E+2(3) 1.6E+1 2.50E+2(2) 3.6E+0 2.38E+2(1) 3.2E+0
G15 40 3.95 1.19E+4(2) 4.8E+2 1.17E+4(1) 6.3E+2 1.23E+4(3) 5.6E+2
G110 20 6.04 ∗(3) ∗9.19E+4(2) 4.1E+3 8.94E+4(1) 4.23E+3
G130 10 0.65 1.75E+2(3) 7.9E+0 1.61E+2(1) 2.9E+0 1.68E+2(2) 2.2E+0
G115 20 0.67 3.54E+2(2) 1.0E+1 3.33E+2(1) 4.5E+0 3.60E+2(3) 4.1E+0
G130 10 1.34 4.20E+2(3) 5.8E+1 3.68E+2(1) 8.5E+0 4.02E+2(2) 2.3E+0
G115 20 5.98 ∗(3) ∗7.83E+4(2) 2.1E+3 7.02E+4(1) 1.9E+3
G117 23 1.71 5.24E+2(3) 5.4E+1 3.43E+2(1) 1.5E+1 4.58E+2(2) 4E+0
G120 20 0.58 2.73E+2(2) 5.5E+0 2.60E+2(1) 3.7E+0 2.86E+2(3) 2.3E+0
G120 20 0.97 2.39E+2(3) 2.1E+1 1.65E+2(2) 6.3E+0 1.45E+2(1) 1.3E+0
G120 20 0.967 4.35E+2(3) 3.1E+1 3.78E+2(1) 5.9E+0 4.08E+2(2) 4.5E+0
G115 30 2.16 1.66E+3(3) 1.8E+2 1.34E+3(1) 3.6E+1 1.41E+3(2) 2.1E+1
Rank value 73 - 41 - 41 -
1 2 3 4 5 6 7 8 9 1011 12 13 1415 16 1718 19 2021 22 2324 25 2627
benchmark instances in Group 1
0
1
2
3
4
5
6
7
8
9
10
11
12
f
104
MA-MLS
MA-OLS
EDA
ILS
AC-ACO
NDE
Figure 3. Comparison of experimental results on the Group 1.
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Table 5. Comparison (I) of experimental results on the Group 2.
Instance MA-MLS MA-OLS EDA NDE
Mean Std Mean Std Mean Std Mean Std
G240 10 0.67 2.34E+2(2) 3.4E+0 2.39E+2(3) 1.3E+0 2.56E+2(4) 3.8E+0 2.29E+2(1) 2.4E+0
G240 15 0.67 2.99E+2(1) 3.1E+0 3.05E+2(2) 2.2E+0 3.20E+2(3) 3.7E+0 3.31E+2(4) 1.5E+0
G220 40 3.61 2.21E+4(2) 3.0E+3 5.28E+4(3) 4.6E+3∗(4) ∗4.10E+3(1) 2.2E+1
G230 30 1.04 5.58E+2(2) 8.9E+0 5.80E+2(3) 6.7E+0 6.36E+2(4) 7.5E+0 4.24E+2(1) 1.3E+1
G230 30 1.94 1.37E+3(2) 6.3E+1 1.54E+3(3) 4.3E+1 1.53E+3(3) 8.6E+1 8.86E+2(1) 2.5E+1
G215 60 3.64 1.73E+4(2) 1.3E+3 2.50E+4(4) 1.2E+3 2.08E+4(3) 1.4E+3 5.82E+3(1) 3.6E+2
Rank value 11 - 18 - 21 - 9 -
Table 6. Comparison (II) of experimental results on the Group 2.
Instance ILS AC-ACO NDE
Mean Std Mean Std Mean Std
G240 10 0.67 2.37E+2(2) 7.2E+0 2.27E+2(1) 3.2E+0 2.29E+2(1) 2.4E+0
G240 15 0.67 2.96E+2(2) 1.2E+1 2.80E+2(1) 4.3E+0 3.31E+2(3) 1.5E+0
G220 40 3.61 1.32E+4(3) 4.0E+3 5.53E+3(2) 4.3E+2 4.10E+3(1) 2.2E+1
G230 30 1.04 5.48E+2(3) 4.1E+1 4.66E+2(2) 6.4E+0 4.33E+2(1) 1.3E+1
G230 30 1.94 1.31E+3(3) 7.8E+1 1.10E+3(2) 4.0E+1 8.86E+2(1) 2.5E+1
G215 60 3.64 1.12E+4(3) 1.9E+3 1.01E+4(2) 4.8E+2 5.82E+3(1) 3.6E+2
Rank value 16 - 10 - 8 -
1 2 3 4 5 6
benchmark instances in Group 2
0
1
2
3
4
5
6
f
104
MA-MLS
MA-OLS
EDA
ILS
AC-ACO
NDE
Figure 4. Comparison of experimental results on the Group 2.
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algorithm in other 15 instances. For 17 benchmark instances in Group 3, the rank-orders of the NDE
are 17 and 19, respectively. Compared with other algorithms, NDE has a better performance in Group
3.
Table 7. Comparison (I) of experimental results on the Group 3.
Instance MA-MLS MA-OLS EDA NDE
Mean Std Mean Std Mean Std Mean Std
G380 15 0.76 1.98E+2(2) 2.3E+0 2.00E+2(3) 2.4E+0 2.07E+2(4) 3.5E+0 1.65E+2(1) 2.0E+0
G320 60 1.51 1.01E+3(2) 2.9E+1 1.06E+3(3) 9.6E+0 1.15E+3(4) 1.4E+1 6.76E+2(1) 5.6E+0
G380 20 0.7 3.18E+2(2) 3.1E+0 3.24E+2(3) 2.1E+0 3.35E+2(4) 4.1E+0 2.59E+2(1) 2.9E+0
G380 20 1.12 3.90E+2(2) 6.2E+0 4.11E+2(3) 4.6E+0 4.53E+2(4) 1.1E+1 2.47E+2(1) 3.2E+0
G320 80 4.33 1.25E+5(2) 2.5E+4 2.33E+5(3) 2.0E+4∗(4) ∗1.90E+4(1) 2.11E+2
G360 30 1.14 5.52E+2(2) 9.9E+0 5.91E+2(3) 6.9E+0 6.95E+2(4) 2.2E+1 5.17E+2(1) 4.6E+0
G360 40 0.69 3.91E+2(2) 3.4E+0 3.99E+2(3) 3.0E+0 4.04E+2(4) 4.4E+0 3.20E+2(1) 2.9E+0
G340 60 1.54 9.48E+2(2) 1.7E+1 9.84E+2(3) 8.9E+0 1.05E+3(4) 1.5E+1 5.37E+2(1) 1.0E+1
G380 40 0.97 4.64E+2(2) 5.4E+0 4.79E+2(3) 3.9E+0 5.01E+2(4) 8.5E+0 3.10E+2(1) 5.2E+0
G340 80 1.05 6.42E+2(2) 8.1E+0 6.40E+2(3) 5.0E+0 6.60E+2(4) 9.5E+0 4.60E+2(1) 4.5E+0
G380 40 2.25 6.54E+3(3) 1.0E+3 1.01E+4(4) 8.8E+2 3.69E+3(2) 3.4E+2 1.42E+3(1) 4.0E+1
G360 60 0.92 4.29E+2(2) 5.3E+0 4.40E+2(4) 3.3E+0 4.36E+2(3) 6.2E+0 2.79E+2(1) 4.0E+0
G360 60 0.922 5.51E+2(2) 6.3E+0 5.64E+2(3) 4.8E+0 5.56E+2(2) 5.8E+0 3.85E+2(1) 7.0E+0
G3120 30 1.2 4.44E+2(2) 6.1E+0 4.64E+2(3) 4.9E+0 4.94E+2(4) 1.2E+1 2.65E+2(1) 6.9E+0
G380 60 0.72 4.18E+2(3) 4.1E+0 4.28E+2(4) 3.9E+0 4.17E+2(2) 4.7E+0 3.17E+2(1) 3.1E+0
G380 80 0.54 3.23E+2(3) 3.3E+0 3.28E+2(4) 3.5E+0 3.08E+2(2) 2.5E+0 2.50E+2(1) 2.4E+0
G360 120 2.07 3.25E+3(3) 6.5E+1 3.36E+3(4) 4.6E+1 3.09E+3(2) 8.2E+1 1.97E+3(1) 4.5E+1
Rank value 38 - 56 - 57 - 17 -
Table 8. Comparison (II) of experimental results on the Group 3.
Instance ILS AC-ACO NDE
Mean Std Mean Std Mean Std
G380 15 0.76 2.00E+2(3) 1.1E+1 1.70E+2(2) 3.6E+0 1.65E+2(1) 2.0E+0
G320 60 1.51 9.60E+2(3) 2.7E+1 7.53E+2(2) 1.9E+1 6.76E+2(1) 5.6E+0
G380 20 0.7 3.31E+2(3) 1.5E+1 2.86E+2(2) 5.5E+0 2.59E+2(1) 2.9E+0
G380 20 1.12 4.16E+2(3) 2.2E+1 3.10E+2(2) 7.2E+0 2.47E+2(1) 3.2E+0
G320 80 4.33 6.42E+4(3) 5.2E+3 2.21E+4(2) 2.0E+3 1.90E+4(1) 2.11E+2
G360 30 1.14 6.21E+2(3) 3.7E+1 5.08E+2(1) 7.2E+0 5.17E+2(2) 4.6E+0
G360 40 0.69 4.00E+2(3) 1.2E+1 3.40E+2(2) 3.0E+0 3.20E+2(1) 2.9E+0
G340 60 1.54 9.05E+2(3) 2.3E+1 6.82E+2(2) 2.0E+1 5.37E+2(1) 1.0E+1
G380 40 0.97 4.99E+2(3) 2.0E+1 3.86E+2(2) 5.2E+0 3.10E+2(1) 5.2E+0
G340 80 1.05 6.23E+2(3) 1.7E+1 5.00E+2(2) 6.6E+0 4.60E+2(1) 4.5E+0
G380 40 2.25 1.42E+4(3) 5.0E+3 1.69E+3(2) 6.6E+1 1.42E+3(1) 4.0E+1
G360 60 0.92 4.48E+2(3) 1.8E+1 3.17E+2(2) 4.0E+0 2.79E+2(1) 4.0E+0
G360 60 0.922 5.58E+2(3) 1.5E+1 4.18E+2(2) 6.9E+0 3.85E+2(1) 7.0E+0
G3120 30 1.2 4.97E+2(3) 1.8E+1 3.40E+2(2) 6.7E+0 2.65E+2(1) 6.9E+0
G380 60 0.72 4.51E+2(3) 1.5E+1 3.29E+2(2) 2.8E+0 3.17E+2(1) 3.1E+0
G380 80 0.54 3.54E+2(3) 1.5E+1 2.49E+2(1) 1.8E+0 2.50E+2(2) 2.4E+0
G360 120 2.07 4.02E+3(3) 3.3E+2 2.19E+3(2) 6.9E+1 1.97E+3(1) 4.5E+1
Rank value 51 - 32 - 19 -
NDE and AC-ACO have similar performance in solving benchmark instances in the Group 1; how-
ever NDE has better performance in solving benchmark instances in the Group 2 and the Group 3.
Therefore, it is not difficult to see that the NDE is effective for solving these 50 benchmark instances.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
benchmark instances in Group 3
0
0.5
1
1.5
2
f
105
MA-MLS
MA-OLS
EDA
ILS
AC-ACO
NDE
Figure 5. Comparison of experimental results on the Group 3.
5.2. Experimental results under the bilevel optimization model
In this part, when the bilevel optimization model is taken into account, it is very necessary to
analyse the effect on the total cost caused by the amount of robots. For the 50 benchmark instances in
subsection 5.1., besides the same amount of robots as in literature, some additional robots are added
to perform these tasks. The purpose of the procedure is to understand the optimal amount of robots,
such that the total cost can be minimized. When robots need to be added, the robot with the lowest
efficiency is first chosen since it can lead to the worse case, increase the cost, and lower the efficiency.
The experimental results are shown in Tables 9 to 11. In Tables 9 to 11, Frepresents the leader’s
objective function in the bilevel programming model, frepresents the follower’s objective function,
and Nirepresents the number of robots added to execute tasks.
Since there are multiple robots with different capabilities in the benchmark instances, in line with
the working capability of robots, the state-determined cost of renting (or buying) robots is set as the
value that equals to λ2times the working efficiency of the robots, and the running cost of the robot is
the product of working efficiency and working time. Without any loss of generality, the values of both
λ1and λ2are taken as 1 in the experiment.
Among the 27 benchmark instances in Group 1 in Table 9, based on the bilevel optimization model,
shows that the optimal solution of 9 instances is better than that of the original instances. For example,
for G15 5 1.66, the optimal solution needs to add 3 robots to the original instances. When the number
of robots is increased, the state-determined cost of robots increases, but the time to complete the task
can evidently decrease. As a result, the leader’s objective function is decreased. So the decision maker
can choose the better option.
For the 6 instances in Group 2 in Table 10, we find that on two instances (G220 40 3.61 and
G215 60 3.64), the better solutions are obtained when an appropriate amount of robots are added. For
the benchmark instances in Group 3, from Table 11, one can find that one instance G220 80 4.33 finds
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Table 9. The solutions of bilevel programming model for Group 1.
Instance best solution of bilevel model best solution of original instance
F f NiF f Ni
G15 4 0.39 7.39E+2 1.03E+2 0 7.39E+2 1.03E+2 0
G15 5 1.66 9.54E+2 7.64E+2 3 1.01E+3 1.16E+3 0
G13 10 1.51 4.46E+2 5.50E+2 1 4.77E+2 8.76E+2 0
G13 15 5.03 4.78E+3 1.71E+3 3 1.10E+3 2.75E+4 0
G15 10 0.93 3.34E+3 4.09E+2 0 3.34E+3 4.09E+2 0
G110 5 1.39 4.07E+3 5.65E+2 0 4.07E+3 5.65E+2 0
G15 10 3.67 3.79E+3 2.84E+3 3 5.34E+3 1.01E+4 0
G15 20 4.36 5.62E+3 1.18E+3 1 6.18E+3 2.03E+2 0
G110 10 3.79 7.28E+3 2.56E+3 6 9.37E+3 7.79E+3 0
G111 11 1.28 1.88E+4 2.50E+2 0 1.88E+4 2.50E+2 0
G130 5 0.46 1.19E+4 1.50E+2 0 1.19E+4 1.50E+2 0
G115 10 1.17 5.74E+3 3.25E+2 0 5.74E+3 3.25E+2 0
G110 15 1.3 3.97E+3 6.15E+2 0 3.97E+3 6.15E+2 0
G120 10 0.47 1.32E+4 9.77E+1 0 1.32E+4 9.77E+1 0
G120 10 0.96 7.25E+3 3.31E+2 0 7.25E+3 3.31E+2 0
G120 10 0.94 7.25E+3 2.38E+2 0 7.25E+3 2.38E+2 0
G15 40 3.95 8.61E+3 4.96E+3 2 1.16E+4 1.23E+4 0
G110 20 6.04 3.28E+4 3.81E+4 2 6.37E+4 8.94E+4 0
G130 10 0.65 1.09E+4 1.68E+2 0 1.09E+4 1.68E+2 0
G115 20 0.67 1.00E+4 3.60E+2 0 1.00E+4 3.60E+2 0
G130 10 1.34 1.10E+4 4.02E+2 0 1.10E+4 4.02E+2 0
G115 20 5.98 2.70E+4 1.78E+4 5 7.23E+4 7.02E+4 0
G117 23 1.71 1.61E+5 4.58E+2 0 1.61E+5 4.58E+2 0
G120 20 0.58 1.29E+4 2.86E+2 0 1.29E+4 2.86E+2 0
G120 20 0.97 2.39E+5 1.45E+2 0 2.39E+5 1.45E+2 0
G120 20 0.967 1.39E+4 4.08E+2 0 1.39E+4 4.08E+2 0
G115 30 2.16 6.33E+3 1.41E+3 0 6.33E+3 1.41E+3 0
Table 10. The solutions of bilevel programming model for Group 2.
Instance best solution of bilevel model best solution of original instance
F f NiF f Ni
G240 10 0.67 1.41E+4 2.29E+2 0 1.41E+4 2.29E+2 0
G240 15 0.67 1.49E+4 3.31E+2 0 1.49E+4 3.31E+2 0
G220 40 3.61 1.27E+4 2.97E+3 2 1.42E+4 4.10E+3 0
G230 30 1.04 1.15E+4 4.33E+2 0 1.15E+4 4.33E+2 0
G230 30 1.94 1.19E+4 8.86E+2 0 1.19E+4 8.86E+2 0
G215 60 3.64 1.12E+4 4.38E+3 2 1.19E+4 5.82E+3 0
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Table 11. The solutions of bilevel programming model for Group 3.
Instance best solution of bilevel model best solution of original instance
F f NiF f Ni
G380 15 0.76 2.89E+4 1.65E+2 0 2.89E+4 1.65E+2 0
G320 60 1.51 1.49E+4 6.76E+2 0 1.49E+4 6.76E+2 0
G380 20 0.7 2.85E+4 2.59E+2 0 2.85E+4 2.59E+2 0
G380 20 1.12 2.81E+4 2.47E+2 0 2.81E+4 2.47E+2 0
G320 80 4.33 2.74E+4 1.37E+4 2 4.46E+3 1.90E+4 0
G360 30 1.14 3.35E+4 5.17E+2 0 3.35E+4 5.17E+2 0
G360 40 0.69 2.14E+4 3.20E+2 0 2.14E+4 3.20E+2 0
G340 60 1.54 2.79E+4 5.37E+2 0 2.79E+4 5.37E+2 0
G380 40 0.97 2.83E+4 3.10E+2 0 2.83E+4 3.10E+2 0
G340 80 1.05 2.78E+4 4.60E+2 0 2.78E+4 4.60E+2 0
G380 40 2.25 3.45E+4 1.42E+3 0 3.45E+4 1.42E+3 0
G360 60 0.92 2.38E+4 2.79E+2 0 2.38E+4 2.79E+2 0
G360 60 0.922 2.21E+4 3.85E+2 0 2.21E+4 3.85E+2 0
G3120 30 1.2 4.11E+4 2.65E+2 0 4.11E+4 2.65E+2 0
G380 60 0.72 3.01E+4 3.17E+2 0 3.01E+4 3.17E+2 0
G380 80 0.54 5.42E+4 2.50E+2 0 5.42E+4 2.50E+2 0
the better solution than original case.
6. Discussion
In the field of task scheduling and route optimization, the multi-point dynamic aggregation problem
is interesting but computationally complex. In the present research, in order to reduce decision-making
cost, some key parameters are predetermined, which should be further optimized, such as the number
of robots as well as tasks. Based on these assumptions, some simplified models have be given, which
simply focus on optimizing the routes of robots to minimize the completion time of all tasks. In fact,
from the decision maker’s point of view, at least two additional costs, robots and property damage, need
to be taken into account. It follows that optimization procedure should be divided into two levels. One
is to determine the number of robots(leader’s level), whereas the other is to optimize the routes of these
robots(follower’s level). As a result, a bilvel programming problem is proposed in the manuscript.
When the leader’s variables are given, the problem can degrade into a single-level problem just as in
literature. Hence, the proposed bilevel problem is more universal than the existing models.
Bilevel programming problems are computationally hard. In order to efficiently deal with the hierar-
chical optimization problem, a two-level DE is developed. In the algorithm for the follower’s program,
a matrix encoding scheme is provided and followed by a newly designed decoding rule. The individual
representation is in nature binary, which is simpler and more efficient than ever. In addition, the mean
location of multiple high-qualify individuals can provide a stabler evolutionary direction than a single
elite individual.
In spite of the fact that the bilevel model is more suitable for practice, because of the discrete vari-
ables and hierarchical structure involved, this problem is still very difficult to solve. More specifically,
for large-scale cases in which too many robots and tasks are involved, the proposed algorithm still
needs to pay expensive computation cost.
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7. Conclusion
In the real-world field, MPDAP is a challenging and difficult optimization problem. In this
manuscript, the problem is re-modeled as a bilevel programming model by considering the total cost of
completion tasks and the damages. In order to efficiently deal with the bilevel programming problem,
a novel differential evolution with a nested structure is developed, in which a new matrix encoding
method is designed to represent the routes of robots. Meanwhile, in order to solve the follower’s prob-
lem efficiently, we also present two-strategy mutation and selection operators. From the experimental
results, one can find that the proposed algorithm has better performance when compared with other
similar algorithms. Also, based on the bilevel optimization model, the solutions with smaller cost are
obtained by dispatching additional robots.
When some interesting topics, such as the dynamic task number and the priority of tasks, are taken
into the model, the optimization procedure may become more complex than ever. In the future study,
more efficient heuristical operators as well as optimization techniques need to be further developed for
this kind of problems.
Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgments
This work is supported by the National Science Foundation of China (No.61966030).
Conflict of interest
The authors declare there is no conflict of interest.
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