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*For correspondence. (e-mail:
[email protected])
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ACKNOWLEDGEMENTS. We thank the Indian Council of Agricul- tural
Research (ICAR), New Delhi for financial support (ICAR- National
Fellow project) and the Director, Indian Institute of Soil and
Water Conservation for providing the necessary facilities and
support to carry out the field experiment and other research work.
Laboratory support extended by the Centre for Advanced Research in
Environmen- tal Radioactivity (CARER), Mangalore University, for
carrying out 137Cs concentration measurements in the soil is
acknowledged. Received 21 February 2019; revised accepted 3 July
2019 doi: 10.18520/cs/v117/i5/865-871
Comparative implementation of the benchmark Dejong 5 function using
flower pollination algorithm and the African buffalo optimization
Julius Beneoluchi Odili1,* and A. Noraziah2,3 1Department of
Mathematical Sciences, Anchor University Lagos, Ipaja, Lagos,
Nigeria 2Faculty of Computer Systems and Software Engineering,
Universiti Malaysia Pahang, Kuantan 26300, Malaysia 3IBM Centre of
Excellence, Universiti Malaysia Pahang, Kuantan 26300, Malaysia
This communication presents experimental research findings on the
application of the flower pollination algorithm (FPA) and the
African buffalo optimization (ABO) to implement the complex and
fairly popular benchmark Dejong 5 function. The study aims to un-
ravel the untapped potential of FPA and the ABO in providing good
solutions to optimization problems. In addition, it explores the
Dejong 5 function with the hope of attracting the attention of the
research com- munity to evaluate the capacity of the two compara-
tive algorithms as well as the Dejong 5 function. We conclude from
this study that in implementing FPA and ABO for solving the
benchmark Dejong 5 prob- lem, a population of 10 search agents and
using 1000 iterations can produce effective and efficient out-
comes. Keywords: Benchmark, comparative implementation, iteration,
optimization algorithms, search agents, test functions. K. A.
DEJONG has made significant contributions to com- puter science.
One of his remarkable contributions is the
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Figure 1. Dejong 1 (sphere) function3. introduction of a set of
functions, sometimes called by his name. In his Ph D thesis, Dejong
introduced five func- tions1,2. These are sometimes called Dejong
1–5, or they are named separately. For instance, Dejong 1 is
popularly called sphere function; Dejong 2 is Rosenbrock; Dejong 3
is step function; Dejong 4 is quartic function, and the complex and
unpopular Dejong 5 is also called Shekel’s foxholes function3. In
computer science and applied mathematics, in gen- eral, test
functions, otherwise referred to as artificial landscapes, are used
in evaluating the efficiency and effectiveness of optimization
algorithms in terms of their robustness, convergence rate,
versatility, precision and general performance. In recognition of
the need for these artificial landscapes, De Jong suggested the
above-listed five functions because of his belief that they
represent the most commonly observed difficulties encountered in
op- timization problems. A brief description of these five
functions is presented below: • The sphere is the most popular of
all of Dejong func-
tions. It is a unimodal, symmetric and smooth func- tion. The
capacity of an algorithm to identify its global optimum is,
doubtless, a good measure of its effectiveness.
• Unlike the ‘popular and easy’ sphere function, the Rosenbrock
function is sometimes considered the nightmare of optimization
algorithms. This function has a very narrow ridge whose tip is
rather sharp, and the Rosenbrock tends to run around a parabola.
Many optimization algorithms find it difficult to discover the
global optimum of the Rosenbrock func- tion.
• The step function is a representation of problem land- scapes
with deceptive flat surfaces. Usually, flat sur- faces pose
problems to optimization algorithms
because they provide insufficient or no information to these
algorithms with regard to which direction is favourable. Unless the
optimization algorithm has in-built effective use of randomness, it
may likely get stuck in one of the flat surfaces.
• The quartic function, on the other hand, is a unimodal function
that is padded with Gaussian noise. The addi- tion of Gaussian
noise makes it extremely difficult for the algorithm to obtain an
appropriate value on any given point during the search process.
This function provides a good test for any optimization algorithm
searching a noisy landscape.
• Finally, the shekel’s foxholes function provides a good example
of multimodal search landscape. This function has 25 local optima
and 1 global optimum. The capacity of an optimization algorithm to
discover each of these 25 local optima as well as locate the global
optimum is a mark of a good optimization search algorithm.
In this study, we focus on Dejong 5 function because of its
complexity. Due to its nature, we prefer to refer to this function
as the multi-legged table function4. It has 25 local optima and one
global minimum. Also, because of its deceptive nature, it poses a
great problem to optimiza- tion algorithms, that researchers tend
to ignore it. This is our motivation as it provides a good test for
optimization search algorithms. Figures 1–5 present the five bench-
mark Dejong functions5,6. In this study, we evaluate the capacity
of the flower pollination algorithm (FPA), one of the less popular,
yet powerful optimization algorithms developed by Yang7 and the
African buffalo optimization (ABO), a recently designed
optimization algorithm inspired by the migrant lifestyle of African
buffalos8,9, to solve the Dejong 5. The choice of these two
algorithms for a comparative study of
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Figure 2. Dejong 2 (Rosenbrock) function5.
Figure 3. Dejong 3 (step) function6.
Figure 4. Dejong four (quartic) function.
this benchmark function is in recognition of their effec-
tiveness10–12 and the need to popularize them among researchers.
The FPA was developed by Yang7 with inspiration from the
pollination characteristics of flowers. Usually flowers are self-
or cross-pollinated. Self-pollinated flow- ers use wind or rain to
help them transport pollen grains for pollination. On the other
hand, cross-pollinators use animals, insects or other agents for
pollination. In the FPA, self-pollinators are deemed to be local
search (exploitation), while the cross-pollinators are regarded as
global search (exploration). An interplay of the exploita- tion and
exploration stages enables the algorithm to arrive at good
solutions12.
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CURRENT SCIENCE, VOL. 117, NO. 5, 10 SEPTEMBER 2019 874
Figure 5. Dejong 5 (shekel’s foxholes) function. 1. Begin 2.
Randomly initialize the flowers as well as the pollinators 3.
Evaluate the flowers and pollinators 4. Ascertain the present best
flower 5. Determine the search constancy probability p ∈ [0, 1]. 6.
While (until termination), 7. For j = 1 to n (n = number of flowers
in the population) 8. If rand < p, (rand=random number) then 9.
Use a Levy flight to ascertain the global pollinator 10. Choose j
and k among the obtained solutions and execute local pollination
(search) with eq. (1) 11. End if 12. Ascertain new solutions and
evaluate the present flower fitness 13. If new solutions are better
than the previous, then 14. Replace the previous with the new
solution 15. Else retain the previous solution 16. End if 17. End
for 18. End while 19. Output best outcome 20. End
Figure 6. Pseudocode of the flower pollination algorithm.
In the FPA, global and local pollination (search) pro- cesses are
controlled by a switch p that has a probability7 p ∈ [0, 1]. (1)
During the global search, flower constancy is represented by xi(t +
1) = xi(t) + L(xi(t) – g*), (2)
where g* represents the present best solution, xi(t) denotes pollen
i at iteration t along a given vector xi and L represents Levy
flight. Since its development, the FPA has been successfully
applied to solve global optimization problems7, load fre- quency
control for a hydro-thermal deregulated power sys- tems13, fractal
image compression, Sudoku puzzle, disc brake design problems,
retinal vessel segmentation, multi- objective problems, disc brake
design problem14, etc. with good results. Figure 6 presents the
pseudo code of the FPA7. The ABO was developed by Odili and Kahar
in 2015 with inspiration from the movement of African buffalos in
their continental landscapes in search of pastures. The African
buffalos manage their large herds sometimes numbering more than
1000 animals using two major sounds:/maaa/calling the animals
together for a diligent search of the landscape since it holds
promise of good pastures, and the /waaa/ calls that caution the
animals to get out of a particular location to explore other gazing
locations. With judicious use of the /maaa/ (exploit) and /waaa/
(explore) calls, ABO is able to arrive at a good and optimized
search results15. Figure 7 presents the pseudocode of the ABO. So
far, the ABO has been successfully applied to solve energy and
delay routing in mobile ad-hoc networks16, symmetric TSP17,
asymmetric TSP18, benchmark global optimization functions19,
numerical function optimization problems, strategic management,
collision avoidance in electric fish, tuning of PID parameters of
automatic voltage regulators9, etc. with good results.
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In this comparative evaluation of FPA and ABO, the emphasis is on
the effect of iteration numbers and search population in obtaining
good results. The experiments were implemented using MATLAB on a
desktop PC (Intel Duo Core i7 370 CPU @ 3.40 GHz, 3.40 GHz, 4GB
RAM, running Windows 10). To ensure fair comparison, both
algorithms were run in the same platform and using the same
language (MATLAB), same population (10 and 50) and same number of
iterations (10, 100, 1000, 5000, 10,000). Also, both algorithms
were used to execute five times each the benchmark Dejong 5
function
1 25
f x j x a
−
= =
= + + −
∑ ∑
ija = 32 16 0 16 32 32 0 16 32
, 32 32 32 32 32 16 32 32 32 − − − − − − − − −
… …
–65,536 ≤ xi = 65.536, i = 1, 2,
* 31.97833.ix ≈ − Other parameters used for the FPA are n = 1, p =
1, upper bound (UB) = 2, lower bound (LB) = –2 and d = 3. For the
ABO, lp1 = 0.7, lp2 = 0.5. The optimal solution of the benchmark
Dejong 5 function is
f (x) = 0.9980.
Table 1 provides details of the experimental outcomes of both
algorithms when the population of buffalos or flow- ers is 10, and
the number of iterations ranges from low (10,100) to medium (1000)
and high (5000, 10,000). Ta- ble 2 presents the experimental
outcome of deploying a population of 50 buffalos or flowers, and
varying the number of iterations from 10 to 10,000. A close look at
Table 1 shows that both algorithms performed well while searching
the solution space with a population of 10 flowers/buffalos.
Nonetheless, when the iteration number was 10, the ABO showed
better results with an average of 12.827. In fact, one run produced
an outcome of 3.3325. This is significant because the optim- al
solution is 0.9980. The result of the FPA was not as good. It
posted an average of 29.980 way off the mark. However, in terms of
execution time, the FPA spent an average of 0.026 sec to 0.112 sec
in case of the ABO. This shows that the FPA executed faster.
Similarly, when the iteration number was increased to 100, output
of the FPA improved significantly to an aver-
age of 1.991 compared to 6.814 of the ABO. The execu- tion time of
the FPA averaged at 0.137 compared to 0.573 for the ABO: another
proof of execution efficiency of the FPA. Similar trend was
observed when the itera- tion count was adjusted to 1000 with the
FPA posting an average result of 1.957 compared to 5.481 for the
ABO. Nevertheless, effectiveness of the ABO was visible in the
algorithm obtaining the optimum solution here (light grey shaded
portions in Tables 1 and 2). In terms of execution time, the FPA
was still faster with an average of 1.137 sec compared to 5.520 sec
for the ABO. With 5000 iterations, the trend was similar as in 1000
iterations. However, remarkable changes were observed when the
iteration was adjusted to 10,000. The performance of the ABO
improved significantly with three optimal solutions and an average
of 0.9981. The FPA however, reversed its excellent run of good
results with two optimal solutions and an average of 0.9984. In
view of the above, we can conclude that the FPA has faster
execution time than the ABO when using a popula- tion of 10
flowers/buffalos. In any case, the ability of the ABO to obtain
optimal or near-optimal solution even with as little as 1000
iterations is not in doubt. These findings are in agreement with
the ‘no free lunch (NFL) Theorem’20 which states that a parameter
of interest to a researcher determines his choice of an
optimization algo- rithm. For this test function, if speed is a
more crucial requirement, then the FPA is recommended; but if
obtain- ing the optimal solution as fast as possible is the primary
concern, the ABO has an edge. Next we examine the performance of
the algorithms when we deploy 50 search agents (buffalos or
flowers). Table 2 presents the experimental outcome. The behaviour
of the algorithms in Table 2 is not much different from the trend
displayed when we deployed 10 1. Begin 2. Initialize the buffalos
randomly to different locations within the solution space; 3. While
(until termination) 4. For j = 1: n (n = the population of
buffalos) 5. Ascertain the buffalos exploitation fitness with: 6.
m′k = mk + lp1(bg – wk) + lp2(bpk – wk) 7. where wk = exploration
move; mk = exploitation move; bg = position of the best buffalo; bg
is the herd’s best fitness; lp1 and lp2 denotes learning factors,
bpk = best location of the kth buffalo 8. Update the exploration
fitness of buffalos with: 9. w′k = (wk + mk)/λ. 10. Check if bgmax
is updating? Yes, go to 11. If No in 10 iterations, return to 2 11.
End for 12. End while 13. Output best outcome. 14. End
Figure 7. Pseudocode of the African buffalo optimization.
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CURRENT SCIENCE, VOL. 117, NO. 5, 10 SEPTEMBER 2019 876
search agents (Table 1). The ABO obtained better out- come (average
5.8331, 2.149, 1.248 respectively) while using 10, 100 and 1000
iterations at an average of 0.29, 2.646 and 26.379 sec
respectively. The results (average: 29.980, 3.981, 1.591
respectively) of the FPA were not as good as those of the ABO.
However, at 5000 iterations, the FPA had a better average result of
1.001 compared to 1.1305 for the ABO. Just like in the earlier
experiment (Table 1), the ABO was able to obtain one optimum solu-
tion at 1000 and 5000 iterations, and three optimal solu- tions at
10,000 iterations. In fact, the ABO had an average of 0.9980, which
is the optimum solution at 10,000 iterations. The FPA had only two
optimal solu- tions at 10,000 iterations, with an average of
0.9984. From the analysis, it is obvious that both algorithms show
consistent behaviour when using either 10 or 50 search agents.
While the FPA proved to be faster than the ABO in solving the
benchmark Dejong 5 function, the ABO had a better run in terms of
obtaining the optimal or near-optimal solutions. Table 1.
Performance of African buffalo optimization (ABO) and flower
pollination algorithm (FPA) when using a population of 10
buffalos/flowers
ABO FPA
Iterations Population fmin Time (sec) fmin Time (sec)
10 10 13.9122 0.243 40.3190 0.025 22.3328 0.092 60.1138 0.025
3.3325 0.075 19.2325 0.026 11.8802 0.075 11.2907 0.027 12.6747
0.074 18.9458 0.029 Average 12.827 0.112 29.980 0.026
100 12.6705 0.574 2.5526 0.173 7.8740 0.568 1.0754 0.124 5.7373
0.572 4.3196 0.136 6.6766 0.581 1.0014 0.125 1.1096 0.569 1.0034
0.125 Average 6.814 0.573 1.991 0.137
1000 2.0259 5.528 3.4697 1.146 3.8699 5.539 0.9961 1.126 0.9980
5.521 0.9982 1.143 10.8268 5.488 3.2507 1.134 9.6829 5.524 1.0724
1.134 Average 5.481 5.520 1.957 1.137
5000 4.9505 27.466 0.9986 5.550 2.0291 27.639 0.9983 5.565 0.9980
27.347 0.9981 5.538 2.9821 27.395 1.0036 5.575 0.9992 27.395 1.0042
5.517 Average 2.392 27.448 1.001 5.549
10000 0.9980 54.655 0.9992 11.097 0.9980 54.861 0.9983 11.035
0.9983 55.299 0.9980 11.316 0.9980 55.129 0.9980 11.335 0.9981
54.914 0.9984 11.096 Average 0.9981 54.972 0.9984 11.176
In this study we analysed the complex and unpopular benchmark
Dejong 5 function using the ABO and the FPA with special focus on
the number of populations and iterations required to obtain optimal
or near-optimal solu- tions. The Dejong 5 function has 25 local
optimal and 1 global optimum solution. The complexity of this
bench- mark test function, should be considered as a merit, since
it makes the function a good testbed for optimization search
algorithms. After a number of experimental evaluations, this study
recommends the usage of 10 search agents (buffalos or flowers) with
this test function or any other problems similar to it. This
conclusion arises from the need to minimize the use of computer
resources and save time while not compromising on search
effectiveness. Since the use of computer resources correlates with
time spent on solving a problem21 coupled with the findings of this
study that the use of more search agents does not lead to a
significant improvement in search outcomes, using a larger
population is not recommended. Both algorithms performed better at
10,000 iterations with the ABO obtaining three optimal solutions
when
Table 2. Simulation results using 50 search agents
ABO FPA
Iterations Population fmin Time (sec) fmin Time (sec)
10 50 7.8742 0.295 40.3190 0.025 3.3238 0.302 60.1138 0.025 1.2178
0.291 19.2325 0.026 4.9463 0.292 11.2907 0.027 11.8032 0.294
18.9458 0.029 Average 5.8331 0.295 29.980 0.026
100 3.4537 2.649 2.5526 0.173 2.0037 2.664 1.0754 0.124 3.2031
2.634 4.3196 0.136 1.0819 2.648 1.0014 0.125 1.0013 2.636 1.0034
0.125 Average 2.149 2.646 3.981 1.366
1000 1.2515 26.554 3.4697 1.146 0.9980 26.309 0.9961 1.126 1.9921
26.248 0.9982 1.143 0.9981 26.469 3.2507 1.134 0.9980 26.315 1.0724
1.134 Average 1.248 26.379 1.597 1.137
5000 1.0882 131.066 0.9986 5.550 0.9980 131.170 0.9983 5.565 1.0106
131.159 0.9981 5.538 2.4279 131.407 1.0036 5.575 0.9987 130.650
1.0042 5.517 Average 1.1305 131.090 1.0001 5.549
10000 0.9981 261.595 0.9992 11.097 0.9980 262.437 0.9983 11.035
0.9981 262.438 0.9980 11.316 0.9980 262.674 0.9980 11.335 0.9980
254.971 0.9984 11.096 Average 0.9980 260.823 0.9984 11.758
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CURRENT SCIENCE, VOL. 117, NO. 5, 10 SEPTEMBER 2019 877
either using 10 or 50 search agents, while the FPA ob- tained two
optimal solutions each on both counts. How- ever, this study
recommends the use of 1000 iterations in order to save CPU
processing time. The results obtained when using 1000 iterations
are close enough to the opti- mum to merit this recommendation.
Indeed, the ABO was able to obtain the optimum solution while using
10 or 50 buffalos in the search space at 1000 iterations. Finally,
in agreement with the NFL theorem, this study concludes that if
speed is the main consideration, then the FPA is recommended.
However, if the primary considera- tion for solving this problem is
the need to obtain the optimum solution, then the ABO is
recommended. Conflict of interest: The authors declare that there
is no conflict of interest.
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ACKNOWLEDGEMENTS. We thank the Department of Mathemati- cal
Sciences, Anchor University Lagos, Nigeria, and the Faculty of
Computer Systems and Software Engineering, Universiti Malaysia
Pahang, Kuantan, Malaysia for providing funds (Grant GRS 1403118).
Received 11 May 2017; revised accepted 20 June 2019 doi:
10.18520/cs/v117/i5/871-877
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