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Towards ASP-based Scheduling for Industrial Transport Vehicles

  • Tesla Manufacutring Brandenburg SE


The increasing number of robots and autonomous vehicles involved in logistics applications leads to new challenges to face for the community of Artificial Intelligence. Web-shop giants, like Amazon or Alibaba for instance, brought this problem to a new level, with huge warehouses and an huge number of orders to deliver with strict deadlines. Coordinating and scheduling such high quantity of tasks over a fleet of autonomous robots is a really complex problem: neither simple imperative greedy algorithms, which compromises over the quality of the solution, nor precise enumeration techniques, which compromises over the solving time, are no more feasible to tackle such problems. In this work, we use Answer Set Programming to tackle real-world logistics problems , involving both dynamic task assignment and planning, at the BMW Group and Incubed IT. Different strategies are tried, and compared to the original imperative approach.
Towards ASP-based Scheduling for Industrial Transport Vehicles *
Felicitas Fabricius
Marco De Bortoli
Gerald Steinbauer
Graz University of Technology
Maximilian Selmair
BMW Group
Michael Reip
Incubed IT
Martin Gebser
at Klagenfurt
The increasing number of robots and autonomous
vehicles involved in logistics applications leads to
new challenges to face for the community of Arti-
ficial Intelligence. Web-shop giants, like Amazon
or Alibaba for instance, brought this problem to a
new level, with huge warehouses and a huge num-
ber of orders to deliver with strict deadlines. Coor-
dinating and scheduling such high quantity of tasks
over a fleet of autonomous robots is a really com-
plex problem: neither simple imperative greedy al-
gorithms, which compromises over the quality of the
solution, nor precise enumeration techniques, which
make compromises over the solving time, are any-
more feasible to tackle such problems. In this work,
we use Answer Set Programming to tackle real-world
logistics problems, involving both dynamic task as-
signment and planning, at the BMW Group and In-
cubed IT. Different strategies are tried, and com-
pared to the original imperative approach.
1. Introduction
Industry 4.0 is bringing more and more interest
toward the digitalization of all productive stages in
the industrial field. Even before that, we all have
been witnesses of the big impact robotics had in
industry, by the automatization of repetitive tasks.
In the last years, thanks to the increasing computa-
tional power, Artificial Intelligence (AI) is spreading
as well, leading to the next step of the integration
*This paper appeared at the 2020 Joint Austrian Computer
Vision and Robotics Workshop.
between robots and production: the automatization
of complex tasks requiring reasoning. In this per-
spective, optimization of logistics is crucial for large
companies, in order to save both time and money.
Still we are dealing with a production environment
which considers a fleet of robots floating around, ef-
ficiently performing tasks and carrying goods where
to model such NP-hard domains a high number of
constraints is needed. For this reason, a imperative
approach become more and more difficult to main-
tain, and cannot benefit from the numerous meta-
heuristics and optimizations (if not manually imple-
mented) already encoded inside the solvers of other
programming paradigms, like declarative program-
ming. Answer Set Programming (ASP) is a fast and
intuitive logic language, which already has many ap-
plications both in industry and in research (see Sec-
tion 2). In this paper we are going to investigate the
difference between the two paradigms, by replacing
the imperative part of the task schedulers used by two
companies, the BMW Group and IncubedIT, with an
ASP implementation. While classical languages are
well suited for greedy algorithms implementation,
declarative programming has other advantages: first
of all, the focus is on the description of the problem,
leaving all the solving details to the external solver.
Moreover, most solvers are configurable with a lot
of meta-heuristics to cut the search space: the user
has only to find the one which fits the problem better,
without implementing anything. Then, since logic
languages are basically based on enumeration tech-
niques, an ASP solver looks for the best solution,
or at least the best one in a given time. Depending
on the size of the instance, this behaviour leads to
huge computational time with respect to a greedy al-
gorithm. We are interested into the analysis of this
trade-off between greedy solving time and declara-
tive solution quality. This paper is based on Felicitas
Fabricius’ Master thesis [6].
2. Related Research and ASP Foundations
The demand for increased complexity and scala-
bility in industry automatization requires more and
more powerful techniques and algorithms. Impera-
tive programming is suitable to write a very problem-
specific solution. However, the development of such
kind of code can be really arduous, time expensive
and difficult to maintain. Optimal task scheduling
and planning, enriched by domain-based heuristics,
requires a huge amount of code lines if written with
an imperative language [17].
Answer Set Programming, and logic program-
ming in general, allows to tackle combinatorial prob-
lem in a very intuitive way, splitting the work into
two phases: the description of the problem and its ef-
ficient solving procedure [7]. The programmer has
only to care of the former, and this requires just
a fragment of the effort required by an imperative
language. Then, (s)he can use one of the solvers
available in the market, like Clingo or DLV, to find
the optimal solution, improving it with a large set
of meta-heuristics. Although the most common ap-
proach is the imperative one, many use-cases of ASP
applied to industry can be found in the literature: in
2017, Dodaro and Maratea designed a shift plan for
164 Italian nurses, calculating the optimal plan for an
entire year in just 50 minutes, using the state-of-the-
art solver Clingo [3]. Staying in the shift scheduling
field, the DLV solver was deployed to find the opti-
mal shift plan for seaport workers [14]. In this case,
the problem was complicated by the fact that the em-
ployees have different qualifications, and there were
different kinds of tasks. Finding of the optimal one
month-long plan required 8 minutes. A similar work,
considering different demands as well, is described
in [2]. Moving to other kind of industrial fields,
ASP was used in E-tourism in order to find the travel
which suits the user the most [1]. In [12] the au-
thors used ASP for phone routing in call centers. The
customer was classified in a category and assigned
directly to the human operator. We can find plenty
of ASP applications regarding task assignment and
routing as well. Examples can be found in [4], [16],
[10], [13] and [15].
Answer Set Programming is based on the stable
model semantics, presented by Gelfond and Lifschitz
in [11] for dealing with logic programs with nega-
tion as failure. With the following we give a quick
overview of the language semantics [2, 7].
Arule rin a logic program is an expression of the
ha1, . . . , am,¬am+1,...,¬an(1)
where a1, . . . , anare atoms of the form s(t1, . . . , tk),
in which sis a predicate symbol and t1, . . . , tkare
terms, viz. constants, variables, or functions, and ¬
stands for default negation. The head hof ris ei-
ther an atom a, a choice {a}, or the special sym-
bol . If his an atom and n= 0, we call rafact,
achoice rule if his {a}, and an integrity constraint
if his ; we skip or , respectively, when writ-
ing rules (1) with n= 0 and integrity constraints.
Alogic program Pis a set of rules and constraints.
In the first-order case, terms occurring in Pmay
include arithmetic expressions, and atoms may be
based on relational operators like “<”. On the other
hand, a term, atom, rule, constraint, or program is
ground if it does not include variables, arithmetic ex-
pressions, or relational operators. A first-order pro-
gram Pstands for the set grd(P)of all instances
of rules and constraints constructible by substituting
ground terms for variables and evaluating arithmetic
expressions as well as relational operators in the stan-
dard way. For details on ground instantiation, we
refer the interested reader to [5, 9]. The semantics
of a logic program Pis given by its stable models,
which are particular sets of (true) ground atoms as
defined in the following. The reduct PXrelative to
a set Xof ground atoms is the set of all rules and
constraints in grd(P)such that {a1, . . . , am} ⊆ X,
{am+1, . . . , an} ∩ X=, and aXif h={a}is a
choice for a rule (1). Then, Xis a stable model of P
if it is -minimal among the sets of ground atoms
such that, for all rules in PX,{a1, . . . , am} ⊆ X
implies hXor aXif h={a}. In addition to
rules, a logic program can contain #minimize state-
ments of the form
#minimize[`1=w1@L1, . . . , `n=wn@Ln].
Besides literals `iand integer weights wifor 1
in, a #minimize statement includes integers Li
providing priority levels [8]. The #minimize state-
ments in Pdistinguish optimal answer sets of Pin
the following way. For any set Xof atoms and inte-
ger L, let ΣX
Ldenote the sum of weights wisuch that
`i=wi@Loccurs in some #minimize statement
in Pand `iholds w.r.t. X. We also call ΣX
Lthe util-
ity of Xat priority level L. An answer set Xof P
is dominated if there is an answer set Yof Psuch
that ΣY
Land ΣY
L0= ΣX
L0for all L0> L, and
optimal otherwise.
3. ASP and Logistics: Two Cases-Studies
To evaluate ASP in an industrial environment, we
discovered two interesting case-studies. Both are re-
lated to Fleet Management Systems (FMS) - one at
Incubed IT, the other one at the BMW Group. In
both cases, the imperatively described task allocation
strategy was replaced by an ASP-based program.
3.1. The BMW Use Case: Task Assignment and
Charging Management
By the following, the requirements for the FMS
at the BMW Group are described. Here, two ele-
mental decisions have to be made. These are on one
hand the assignment of tasks to the vehicles and on
the other hand the assignment of charging and park-
ing stations to the same vehicles. Both decisions are
made online, which means that neither tasks nor the
needs for charging (and parking) are known before-
hand. With task we mean a transportation job of a
container, accomplished by a vehicle, from a station
to another one. The required time is estimated from
the Euclidean distance.
For the task assignment, the standard C# sched-
uler applies a trivial first-in-first-out (FIFO) strategy,
which means that earlier created tasks have to be ex-
ecuted first. By that, the criterion for the selection
of tasks, formulated as a constraint, is not to assign
a task if there is another appropriate task with earlier
creation time assignable. Vehicles on the field must
have a battery level at a minimum of 25 %, and charg-
ing vehicles a battery level of 40 % to be assigned to
tasks. The optimal assignment of vehicles to tasks
is based on the traveling costs that are set to be the
Euclidean distance between robots and the first goal
of the assigned task. The used optimization criterion
ensures the lowest traveling cost for the tasks with
earliest time of creation.
In ASP a different optimization criterion is used,
in order to achieve a better overall quality of the so-
lution. The Euclidean distance for all assignments is
summed up and minimized, in order to have a better
make-span and save more energy. Considering Tand
Ras the set of tasks and robots respectively, task as-
signment is encoded by the following logic formulas:
tT(|{rR|(assign(t, )}| ≤ 1)
t1, t2T, rR
(assign(t1, r)assign(t2, r)t16=t2⇒ ⊥)
The first formula may (non-deterministically) as-
sign each task to one robot at most. The second one
makes sure that two different tasks are not assigned
to the same robot. The non-deterministic choice is
driven by the optimization algorithm. In ASP, above
formulas are encoded as follows (:- stands for ):
Listing 1 ASP encoding of the task assignment
0{a s s i g n (T , R ) : r o b o t (R , , , ) }1:
t a s k (T , , ) .
:a s s i g n (T , R ) , a s s i g n ( T2 , R ) , T ! = T2 .
The first rule makes use of both a conditional lit-
eral and a cardinality constraint. A conditional literal
a:b1, . . . , bnis a nested implication, where aand
b1, . . . , bncan be seen as the head and the body of a
rule respectively. The cardinality constraint is used
to ensure that each task is assigned to one robot at
most. Given x{head}y:- body, the meaning is that,
for each different body instantiation (for each task
Tin our case), the head is instantiated from xto
ytimes (from 0to 1in our case). In our code this
implies that, for each task T, at most one robot R
is assigned inside the head. The second rule is an
integrity constraint. In the case that after the task
assignment was performed unassigned vehicles are
remaining, these free vehicles are assigned to charg-
ing stations and parking places. The rules used for
this particular assignment problem are defined sepa-
rately for vehicles on the field and vehicles currently
in charging stations. A charging vehicle can only
be assigned to a charging station if the battery level
is below 90 %. Vehicles on the field can be sent to
charging stations any time, regardless of the current
battery level. Charging vehicles can go to a park-
ing place only if the battery level is above or equal
90 %, whereas vehicles on the field can go to parking
places independently from the battery level. In the
original implementation, priority is given to vehicles
with the lowest battery level. Similarly to the FIFO
strategy in task assignment, first we assigne the least
charged vehicle to the closest station, then the second
least charged one, and so on. However, this imple-
mentation shows its limits on circumstances where
multiple robots have critical battery levels that differ
only in a very small amount. For this reason, and
since our goal with the declarative encoding is to im-
prove the overall quality of the assignment, in the
ASP implementation we minimize the overall trav-
elling distance, like we do for the task assignment.
The rules and constraints needed are very similar to
the ones we used before, where PR1, PR2 are some
user-defined parameters required for the assignment:
Listing 2 ASP encoding of the charge assignment -
assignment rule
0{c ha r ge ( S , R) : s t a t i o n ( S , PR1 , PR2 ) }1
:r o b o t ( R , PR1 , P R2 , ) .
3.2. The Incubed IT Use Case
Incubed IT is a robotics company focused on soft-
ware development for smart robots. They typically
deal with problems of the same type as the previ-
ous use-case we just discussed above. Thus the main
topic is multi-robot planning and scheduling. For this
reason, programmers at Incubed IT designed a highly
parameterized platform which, if configured accord-
ingly, can face a lot of different situations, like ware-
houses of online traders, logistic centers of super-
markets and car manufacturing plants. Fortunately,
this platform is quite modular, partially centralized
and partially decentralized, with a main FMS mod-
ule which is responsible for the coordination of the
many parts of the system. Thanks to this design, re-
placing the old solving module with the ASP solver
has been easy to do.
In the imperative implementation, two kinds of
optimization costs can be used: FIFO and global op-
timum. The former does not require more explana-
tions, while the latter considers a priority number as-
sociated to each task. Regarding the task assignment,
we stick to the important constraint rule in ASP: we
can assign only one vehicle to a task, and only one
task to a vehicle at a time. The same rules and con-
straints we used for the BMW use-case thus fit to In-
cubed IT software as well.
We can now focus on the other problem to solve,
the charge assignment. The charging strategy here is
more sophisticated than in the BMW-case, and robot
can be sent to charge for four different reasons: fixed
time slot charging: robots are assigned to charging
stations due to a reached time slot; critical charg-
ing: robots are assigned to charging stations due to a
battery level below the critical charging limit; busy
charging: robots are assigned to charging stations
due to a battery level below the busy charge limit;
idle charging: robots are assigned to a charging sta-
tion due to not enough appropriate assignable tasks.
Obviously, all of these parameters (critical and busy
charging limit, duration of the time slot) can be cus-
tomized by the user. We define now the rules and
constraints used to implement the third situation:
Listing 3 Assignment of busy charging robots
0{c ha rg e ( S , R b us y ) : c h a r g i n g s t a t i o n ( S
, , , , ) , r o b o t s t a t i o n ( R, S ) }1 :
r o b o t c h a r g e o p t ( R , BL , a u t o m a t i c m o d e ,
, , , BCL , CCL) , BL <= BCL, BL >CCL .
Listing 4 Avoidance of double allocations
:c h a r g e ( S , R , ) , c h a r g e ( S , R2 , ) , R! = R2 .
:c h a r g e ( S , R , ) , c h a r g e ( S2 , R , ) , S ! = S2 .
:a s s i g n ( T , R) , c h a rg e ( , R, ) .
In contrast to the BMW case-study, here we do not
handle the two problems of task and charge assign-
ment separately: we optimize two different weighted
criteria. The most important one is the minimization
of the overall travelling distance of robots assigned
to tasks or to charging stations due to forced time-
slot, critical or busy charging. Then, the same op-
timization, with a lower weight, is applied to robots
assigned to parking places and charging stations for
idle charging.
4. Evaluation of Runtime and Quality for
both Case-Studies
In this section, we present a brief evaluation of
both case-studies. We designed several instances
for each case-study involving different numbers of
robots, orders, charging and parking stations to test
different scales. Subsequent, the runtime as well as
the quality of the solutions for these scenarios are
compared. Furthermore, since Clingo can combine
different meta-heuristics and parallelization strate-
gies for the solving process, we tested all the com-
binations between them in order to find, for each
case, the best one. As a result of the evaluation of
these solving approaches [6], we chose the branch-
and-bound-based optimization strategy in combina-
tion with splitting-based search multithreading and
four threads for the two BMW assignment prob-
lems, while for Incubed IT the best approach is the
Vsids Heuristic combined with compete-based mul-
tithreading with four threads.
The systems of BMW and Incubed IT have been
tested on devices with the following specifications.
At BMW an Intel(R) Core(TM) i5 with a 1.70GHz
processor and 8GB RAM is used. At Incubed IT an
Intel(R) Core(TM) i5- 7200U is used with a 2.50GHz
processor and 8GB RAM. On both systems Windows
10 is installed. Clingo is running in version 5.3.0
with Gringo V5.3.0. and Clasp V3.3.4.
4.1. Evaluation at BMW Group
In Table 1, the mean value and the standard de-
viation of the runtimes for all test scenarios (10 for
each scenario) are shown and the number of solved
test runs is given. If the optimal solution is not found
within the BMW-specific time limit of 60 seconds,
the solving process is aborted. Consequently, these
aborted test runs are not considered in the calcula-
tions for the mean and standard deviation. The mean
performance of the imperative method is for every
scenario the best. As shown in the tables, two differ-
ent ways of using ASP were tested. In the first one,
the solver is directly called inside C#, while in the
second we run ASP standalone. The serious perfor-
mance issues of the former indicate potential for an
improved incorporation of the ASP call in C#.
The instances are formed as follow: for the test
scenario 1, we have 5 tasks and 5 robots; for the sce-
nario 2, 20 tasks and 12 robots; finally, scenario 3 has
50 tasks and 30 robots. The positions of the robots
and stations of the tasks are randomly placed on a
1000 m ×1000 m area.
Looking at the results in Table 1 the imperative
solution seems the winner, but in ASP not the Eu-
clidean distance for single robot is optimised, but
the traveling costs of the whole fleet. So, by using
ASP, we are rewarded with far better quality solu-
tions, as witnessed by Table 4, where traveling costs
for scenario 3 are shown. This scenario is particu-
larly interesting, since ASP was not able to find the
provable optimal solutions within the time limit. Al-
though, while looking for that, solvers like Clingo
keep returning the best solution found so far, as soon
as it finds a better one. Looking at Table 4, we can
see that the best ASP solution found within 1 second
considerably beats the C# solution. However, in this
scenario we do not get an improvement with higher
time limits. Results with the other scenarios are sim-
ilar, with the imperative implementation never being
close to the ASP traveling distance. This particular
problem highlights the performance-quality trade-off
between the two approaches.
In Table 2 the mean value and the standard devia-
tion of the runtime of every test scenario is shown,
considering the charge and park problem. Same
rules as before are applied regarding the time limit
of 60 seconds. The instances are formed as follow:
2 charging stations (CS), 3 parking places (PP) and
3 robots (R) for scenario 1; 7 CS, 14 PP and 3 R for
scenario 2; finally, 17 CS, 33 PP and 30 R for sce-
nario 3.
The imperative C# approach shows for all
scenarios a better performance than the ASP-
implementations, which, as in the task assignment
problem, makes use of a different optimization, mini-
mizing the overall travelling distance between robots
and stations, while the C# program prioritizes the
robots with the most critical battery level. In con-
trast to the task assignment, in this case the problem
is too complex to ASP, which does not succeed in
finding good quality solutions (Table 4) and, in some
cases, it does not succeed to find a solution at all.
This observation leads to the assumption that the en-
coding of the park and charge assignment problem in
ASP is not optimal, as the performance of the task
assignment encoding for similarly scaled instances is
significantly better.
4.2. Evaluation at Incubed IT
In the Incubed IT use-case, the two problems, task
assignment and park and charge assignment, are han-
dled together, according to our characterization in the
previous section. In Table 3, the mean value and stan-
dard deviation of the runtimes for the test scenarios
solved with the original code and with the in-Java
integrated ASP are shown, together with standalone
ASP. A timeout is reached when a test run requires
more than 30 seconds to find an optimal solution.
Test runs that reached the timeout are not considered
in the calculation for the mean and the standard de-
viation. The testing environment has a floor area of
100 m ×86 m where the robots are freely movable.
The three scenarios we are going to test are formed
as follows: for scenario 1, we have 5 robots (R),
3 charging stations (CS), 7 parking places (PP) and
5 tasks (T); 10 R, 6 CS, 14 PP and 10 T for sce-
nario 2; finally, for the last scenario we have 30 R,
18 CS, 42 PP and 15 T.
As we would expect from an NP problem solver,
ASP within C#
µ[ms] σ[ms] # TRS µ[ms] σ[ms] # TRS µ[ms] σ[ms] # TRS
1 0.00 0.00 10 415.50 18.16 10 8.30 8.92 10
2 0.30 0.48 10 2,802.20 4,445.21 10 1,428.90 2,746.91 10
3 0.00 0.00 10 / / 0 / / 0
Table 1: Runtime and solved test runs (TRS) for the different BMW task assignment implementations
ASP within C#
µ[ms] σ[ms] # TRS µ[ms] σ[ms] # TRS µ[ms] σ[ms] # TRS
1 0.00 0.00 10 473.80 81.24 10 13.90 8.88 10
2 16.20 4.87 10 788.10 350.75 10 341.70 404.17 10
3 1,753.30 127.58 10 / / 0 / / 0
Table 2: Runtime and solved test runs (TRS) for the different BMW park and charge assignment implementa-
the reader can notice from the results that ASP is
faster than the Java program while solving small
instances, regardless of the number of constraints.
Though, once the size of the problem hits the com-
binatorial blow-up point, it fails to return an optimal
plan within time.
To measure the quality of the solution, we con-
sider the two metrics we described in the previous
section: most important are the overall travelling
costs for task assignment and critical charging; the
travelling distance of the other kinds of assignment
(like for parking places) are then considered. Since
the optimization strategy adopted with ASP is very
similar to the one already used in the original pro-
gram, in all the scenarios in which the optimal declar-
ative solution is found within time, its quality w.r.t.
to these metrics coincides to the Java solution qual-
ity. For this reason, like we did for the BMW case,
when the ASP solver fails to find the optimal solu-
tion within the limit, we are interested in the analysis
of the best ASP solution found so far. This situation
shows up in the third scenario. Looking at Table 4,
where the total cost (which is the weighted sum of the
two metrics) is shown, it can be seen that the original
implementation in Java provides a significantly faster
and better solution than the ASP implementations.
5. Conclusion
The goal of this work is to make a comparison,
in different real-world logistics scenarios, between
the classic imperative paradigms and the declarative
ones. Answer Set Programming was chosen because
of its high efficiency, as witnessed by the many ap-
plications in industry. To achieve that, the FMS of
BMW and Incubed IT were first analyzed, and then
integrated with a new scheduler modeled in ASP. In
the previous section, results and comparison between
the two approaches in both companies are shown and
analyzed. As we expected, there is not a clear winner
between the two systems, but this comparison high-
lighted the pros and cons of both languages, whose
performance highly depend on the kind and size of
tasks to be accomplished. One main quality criterion
of the FMS is the performance and the quality of the
results. To evaluate the criterion, test scenarios have
been set up that are based on typical use-cases of
the FMS. Regarding the BMW use-case, the imper-
ative solution is significantly faster than the declara-
tive one, especially for the task assignment problem.
However, in ASP we make use of a different opti-
mization technique, which rewards with better solu-
tions . This different strategy led to a trade-off be-
tween solving time and solution quality: if the im-
perative method is faster, ASP finds better solutions.
The Incubed IT use-case gave instead different re-
sults, making clear how a very specific scenario can
benefit from a particular approach rather than a gen-
eral one. However, a common behavior can be seen
from both BMW and Incubed IT, which represents
the main weakness of ASP and enumeration tools in
general. It does not scale over the size of the prob-
lem. Yet in the IncubedIT scenario in which ASP
ASP within Java
µ[ms] σ[ms] # TRS µ[ms] σ[ms] # TRS µ[ms] σ[ms] # TRS
1 278.30 226.64 10 121.30 143.50 10 71.90 14.96 10
2 491.30 223.23 10 495.00 339.16 10 278.67 236.67 10
3 2,572.80 4,894.22 10 / / 0 / / 0
Table 3: Runtime and solved test runs (TRS) for the different Incubed IT assignment implementations
Use-case Imperative ASP after 1 sec ASP after 5 sec ASP after 60 sec
BMW task assign. 7242 3834 3834 3834
BMW park & charge assign. 3217 6979 6530 4669
Incubed IT assign. 186 293 302 257
Table 4: Traveling costs for all the use-cases [m]
does not experience a combinatorial blow-up, it finds
the best solution in less time than Java, without com-
promising over the quality.
To conclude, this work has shown that declarative
programming can perform well on real-world logis-
tics scenario, especially when we are interested in the
quality of the optimization. Another important ad-
vantage of this approach is the separation between
the description and the solving of the problem. In
fact, performance can be an issue with ASP, espe-
cially in a dynamic planning scenario, but fortunately
state-of-the-art solvers like Clingo or DLV come with
many meta-heuristics and optimizations to play with.
Once the proper settings for the specific scenario are
found, solving time can improve considerably, with-
out having to modify the code at all. In all the cases
in which a greedy algorithm is proven to perform
well, in terms of both quality and solving time, im-
perative programming still remains the best choice.
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