-The optimization problem is important due to practical use in various areas of our lives. Unfortunately, quite often there is a situation that we need to find the minimum values with certain criteria. Finding a solution using a classic approach is impossible to apply, which is due to an infinite number of solutions. That is why heuristic algorithms are used to find the optimal solution in a finite time. In this paper we propose the idea of manipulating the initial population due to better distribution of individuals in the whole space. The proposed solution has been described, tested and discussed.
Optimization algorithms find great practical use in various areas of our lives. Designing a construction such as a house requires an ideal weight distribution on the columns. From a mathematical point of view, the pressure, length and width of walls or columns can be described with the help of several equations, where the unknown value will depend on all elements. However, finding the best values to make this construction also as cheap as possible is quite a complicated problem. For this purpose, various algorithms are used that allow finding solutions that meet all criteria.
However, the problem is to search for these values. Quite often the set of values is an infinite one. And this prevents the use of iterative algorithms. These types of problems have contributed to the creation of heuristics, ie methods that do not guarantee ideal (and sometimes even correct) solutions in a finite period of time. Although, a large number of optimization problems are possible to solve using them.
intelligence, where training the classifier occurs by minimizing weights in order to get the least error [1], [2] or the optimization used in the operation of various systems [3]–[6].
extremes for a specific function f (x) : Rn ! R for a c 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0) given points x = (x0; x1; : : : ; xn 1), xi 2 hLi; Rii for i = 0; 1; : : : ; n 1. In the case of minimization problem, it can be formulated as
subject to f (x) g(x) Li 0 xi
Ri
i = 0; 1; : : : ; n
1;
(
Cuckoo Serch Algorithm (CSA) is one of the heuristic algorithms based on living creatures – in this case on cuckoos [10]–[12]. After observation of these birds, people noticed, that cuckoos have specific way to hatch their eggs. They actually do not hatch them by themselves, but search another bird’s nests to leave eggs there. In that case, they need to find the nest whose host does not recgnize unfamiliar egg.
The behaviour of these birds inspired to extend minimization algorithm. CSA assumes that cuckoo is a point x = (x0; : : : ; xn 1) on the n-dimension solution space. Of course, modeling the natural life of these creatures is impossible, so it is simplified due to the following points the size of population is constant in all iterations, each cuckoo can lay only one egg per iteration, cuckoo is identified with egg, host may detect unfamiliar egg with some propability, and in that case when egg is found, the cuckoo has to look for new place in the solution space.
L(x; ; ) = r e 2 (x ) 2 p(x )3
formula, the cuckoo’s movement is done using the following equation x = x
L(x; ; ):
The second step is host decision because after the cuckoo
flight, her egg is tossed to the nest. In fact, cuckoos can mask
(
H(x) =
( >
the egg remains in the nest
the egg is removed from the nest
(
fintess function f ( ), the solution space, threshold value
solution space; i := 0; while i t do k = 0; while j k do
if > then
end end end
behavior during hunting [13]. Polar bear must find the prey and if he doesn’t find anything, he must use the ice floak as a way to move a certain distance. He uses it to drift to another location, where possibly he will find the prey (especially seals).
The basis of this algorithm are similar to the CSA, we have
a population of k bears, where each of them can be represented
as point x = (x0; x1; : : : ; xn 1) in some solution space. At the
beginning, an initial population is created with one difference
– only 75% of individuals are generated, the remaining 25%
depend on Arctic conditions and reproduction.
(
Each bear moves only if the new position is better than the current one in relation to the fitness function f ( ). It can be defined as the following equation (xtj )(i) = (xtj 1)(i) + sign (!) + ; where is a random number in (0; 1i, ! is the distance between two spatial coordinates and is a random value in the range of h0; !i. The distance is defined using simple Euclidean metric between two points and described as d (x)(i); (x)(j) (xk)(i)
(xk)(j) 2: vun 1 = tuX
k=0
water. In a situation where the bear flows on the ice floe, he closely observes the surrounding water. In the case of noticing the future victim, the bear dives into the water and attack. There are occasions when a bear very suddenly throws himself on the victim, if there is a chance that he will run away. In the modeling of this phenomenon, the polar bear’s area of view should be considered as
r = 4a cos( 0) sin( 0);
where a 2 h0; 0:3i is a visible distance, 0 2 (0; 2 i is the
angle of approaching to the victim. This equation is used to
describe the local movement of these individuals, where each
spatial coordinate is modified using
8>x00 = x0
>
>>>x01 = x1
>
>
>>>x02 = x2
>
>
>><:::
>>>x0n 2 = xn 2
>
>
>
>
>
>
>
>>>x0n 1 = xn 1
>
:
r cos(
X r sin( k) + r cos( n 1) k=1 "n 2
X r sin( k) + r sin( n 1) k=1 # # where 1; 2; : : : n 1 2 h0; 2 ).
mental to the individuals living there, for this purpose a random value 2 h0; 1i is introduced, which allows to introduction the conditions of freezing or dying from hunger as the following rule (Death
< 0; 25 > 0; 75 where, if the first condition is met, the weakest individual dies (on the condition that the population size is greater than half). In the second case, the two individuals in a given iteration (one is the best one in whole population relative to the fitness function and when the size of population is smaller than assumed) reproduce as (xtj )(new) = (xtj )(best) + (xtj )(i) 2 ; ; where (xt)(best) and (xt)(i) are two individuals selected from the best in whole population.
fintess function f ( ), the solution space
a given solution space; i := 0; while i t do k = 0; while j k do
if new position is better then
end if end end end
if > 0:75 and the population size is correct then
< 0:25 and the population size is correct then
Take the two best individuals for the
reproduction process according to Eq. (
space, i.e. hLi; Rii. Having k individuals, the area can be divided into m parts as
As part of checking the validity of the proposed solution, the test function described in Tab. I were used (with dimension equal to 10). Tests for described algorithms with space manipulation and without were conducted for two different size of population – 100 and 1000 during 300 iterations. Note that the individuals will be placed at random, so to authenticate the results, each algorithm will be made 10 times, and the result averaged. Obtained solutions are presented in Tab. III and II.
every case more precise, which allows us to stalk that it is worth manipulating the solution space and dividing it into smaller subsets in which individuals will move in heuristic algorithms.
Algorithm 2: Polar Bear Algorithm. als in the initial populations in order to increase the search of the full area. The tests were carried out using two algorithms inspired by nature – Cuckoo Search Algorithm and Polar Bear
V. MANIPULATION OF THE INITIAL POPULATION Algorithm. The obtained results indicate a much faster finding The idea is to place individuals at similar distances, which of the extreme, and compared to the same number of iterations has a chance to guarantee a search of a larger area. The reason – the solutions are more accurate. This allows to say that is that the placement of individuals in the original algorithms heuristics can work more effectively when their randomness is based on random choice. This can lead to the situation that is minismized.
CSA with modification Function name
CSA with modification
PBA