A Bio Inspired Algorithm: Emperor Penguin Optimizer

1.Introduction

 The Emperor Penguin Optimizer Algorithm (EPO) is based on the huddling behavior of emperor penguins. The main process of EPO is to generate the huddle boundary, compute temperature around the huddle, calculate the distance, and find the effective mover. These steps are mathematically modeled and implemented on 44 well-known benchmark test functions [1]. Emperor penguins are highly social, living and feeding in large groups which having the strategy of actively aggregate to benefit from the warmth of conspecifics in response to low ambient temperatures. As the emperor penguins have no nest for breeding and no individual territory, they can form dense clusters of thousands of individuals, so-called huddles, which provide them with an effective protection against cold temperatures and wind they can endure the frigid cold of an Antarctic winter, when temperatures plummet to -20 °C or below. To prevent themselves freezing to death, they huddle together in tightly-packed groups to conserve heat and shelter themselves from the intense winds.

Emperor Penguin Optimizer algorithms have been proposed to solve the variety of real-life engineering optimization problems [2]. These problems are divided into various categories whether they are constrained or unconstrained, discrete or continuous, static or dynamic, single or multi-objective. To increase the efficiency and accuracy of these problems, researchers have encouraged to rely on metaheuristic optimization algorithms [3] which  is based upon  the  population meta-heuristic rooted from evolution that are recombination, mutation, and finally selection. The proposed algorithm introduces a mechanism based on the huddle boundary of emperor penguins, which is named the EPO. In that method, the temperature profile around the huddle is calculated, and we calculated the body temperature and body heat radiation of each penguin.

2. Inspiration of Emperor Penguin Optimization Algorithm (EPO)

The emperor penguin scientifically named as Aptenodytes forsteri, is the group of aquatic flightless birds highly adapted for life in the water which is tallest and heaviest among all living penguin species. The emperor penguin typically lives 15 to 20 years in the wild, but some records indicate a maximum lifespan of 40 years. Emperor Penguin Optimizer (EPO) is inspired by social huddling behavior of emperor penguins to survive successfully in the depth of Antarctic winter. The main process of EPO are inspired by huddling behavior which is to determine the huddle boundary of emperor penguins, calculate the temperature profile around the huddle, determine the distance between emperor penguins and effective mover.

 As the emperor penguins have no nest no individual territory, they can form dense clusters of thousands of individuals, so-called huddles, which provide them with an effective protection against cold temperatures and wind [4]. In a huddle, the body surface temperature of the penguins can rise within less than 2 h to 37 • C. Huddling occurs most frequently during breeding in the midst of winter. Huddles are discontinuous events that last for relatively short durations of few hours corresponding to storm events. The number density in a huddle at a colony may be as high as 10birds. Larger huddles are more able to efficiently conserve heat and groups can be made up of several hundred birds, with up to 10 individuals per square meter.  Huddling tends to occur at night, when temperatures are likely to be coldest. Its benefit from huddles because of the reduction of body surface area exposed to the cold and owing to the warm temperature inside the huddle [5]. Despite the fact that huddles achieve these high ambient temperatures, emperor penguins benefit most from the huddle through the reduction of cold-exposed body surface. The distance is comparable to twice the thickness of the compressive feather layer of around 1.2 cm. This suggests that the penguins touch each other only slightly when standing in a huddle, without compressing the feather layer to maximize huddle density without compromising their own insulation.

3. Emperor Penguin Optimization Algorithm (EPO)

 Emperor penguins are the only species that gather to survive during the Antarctic winter. The male and female are similar in size with feathers of the head and back are black and sharply delineated from the white belly, pale-yellow breast and bright-yellow ear patches. Its sustenance consists of fish, crustaceans, krill, cephalopods, and squid through the hunting process [6]. Emperor Penguins have a rough and spiky tongue which helps them when trying to eat slippery fish. Both male and female emperor penguins forage for food up to 500 km from colonies while collecting food to feed chicks. They huddle together to escape wind and conserve warmth. As a defence against the cold, a colony of emperor penguins forms a compact huddle ranging in size from ten to several hundred birds, with each bird leaning forward on a neighbour. Emperor penguins breed in colonies scattered around the Antarctic continent. Most colonies are situated on the fast-ice that is locked between islands or grounded icebergs.

4. Lifecycle of EPO:

            Emperor penguins bred during the cold Antarctic winter, where the temperature can reach -30C and below and it’s begun mating in March or April and is, typically taking one mate per year [7]. Once in pairs, couples form around the colony together, with the female usually following the male the female lays one egg in May or June, transfers the egg to the male, and returns to sea to feed while the male incubates the egg in his brood pouch for about 65 days. Hatching may take as long as two or three days to complete, as the shell of the egg is thick. Newly hatched chicks are covered with only a thin layer of down and entirely dependent on their parents for food and warmth.  After the chick hatches, in august the male sets the chick on his feet and covers it with his pouch, feeding it a white, milky substance produced by a gland in his esophagus. When the female returns from feeding, from September to October the male departs the breeding site to take his turn feeding. A few couple of day later, in November, chicks begin moulting into juvenile plumage, which takes up to two months and is usually not completed by the time they leave the colony. Adults cease feeding them during this time. All birds make the considerably shorter trek to the sea in December and January.Offspring mortality may result from a variety of causes, including dropping the egg during the initial transfer from female to male, insufficient prey availability, and extreme weather. These birds reach their full growth within 3 years, and then they can breed.

5. Steps of EPO

Emperor penguins, huddling is the key to survival during the Antarctic Winter. To conserve energy and protect themselves from cold, they adopt a behavioral strategy of huddling close together in larger groups. Huddling is considered key to their ability to live in such a cold place. The huddling behavior of the emperor penguins is classified into four sections as:

• To generate and determine the huddle boundary of emperor penguins.
• To calculate the temperature profile around the huddle.
• To determine the distance between emperor penguins.
• To relocate the effective mover

5.1. Generate and determine the huddle boundary of emperor penguins

In the phase of huddling, emperor penguins generally position themselves on a polygon shaped structure edges and they have at least two neighbors which is chosen randomly to form the huddle. The wind flow around the huddle is determined to find the huddle boundary around a polygon. However, the wind flow is faster than the movement of an emperor penguin. To generate the huddle boundary of emperor penguins let as defined the wind velocity as H (m) and random number of penguins as N (m).

HB =H (m) + (d)*N (m)

Where HB denotes the huddle boundary and (d) denotes the density function of the polygon edges.

5.2. Calculate the temperature profile around the huddle

Measurements of the body temperature of penguins in various environments and their relation to weather conditions have led to significant insight into huddle formation. To determine the wind flow around the huddle, we need only to consider a two dimensional flow around a polygon [8]. Moreover, we assume that this flow is in viscid and irrotational. The computation of the temperature profile needed to compute the local rate of heat loss. Because the wind flow does not depend on the time elapsed since a penguin has relocated, and is only dependent on the huddle shape. Consequently, we are able to use the mathematics of complex variables and the physical theory of potential flow to describe the flow around the huddle. In mathematically the temperature profile around the huddle T is calculated as;

T= T(x- Max ^iteration)  

Where x= current iteration

                  Temperature T= 0, when the radius is R>1

                  Temperature T=1, when the radius is R<1

5.3. Determine the distance between emperor penguins

The distance between the emperor penguins is calculated after the generation of the huddle boundary to obtain the best optimal solution the best optimal solution is the fitness value of the emperor penguins. The huddle boundary is determined uniquely by connecting the locations of penguins with fewer than six neighbors. Furthermore, we assume that all penguins in the huddle have at least two neighbors, in which case, the area generated by connecting the lattice points on the huddle boundary is a polygon, so that there are no empty spaces within the huddle [9]. We therefore initiate our simulations by generating a huddle satisfying these conditions, starting with five penguins and adding penguins at locations chosen randomly among the eligible positions (adjacent to the huddle, with two neighbors, and leaving no empty space). Once the initial huddle is formed, we consider that the number of penguin within it remains constant. In mathematically the distance between the emperors penguins are calculated as;

H (P) = N (p) + Ѱ Z_n

Where Z_n denotes the random number drawn from the uniform distribution, Ѱ denotes the qualifying the random component related to the best optimal solution and N (p) denotes the number of penguins in the huddle.

5.5. Relocate the effective mover

Among all penguins located at the huddle boundary, we call the penguin with the highest effective heat loss rate, corresponding to the largest value of H(p), the “mover.” The mover vacates its current position and moves to a new position on the huddle boundary where the local heat loss rate is minimal. Because the huddle is situated on a flat plane, the mover can access any location around the huddle, but it cannot displace other penguins [10]. This movement results in a generic motion from the windward side of the huddle to its leeward side. In particular, the mover is relocated to a new position on the huddle boundary with at least two neighbors so that the huddle shape remains a polygon. The first neighbor is chosen as the penguin with the smallest heat loss rate corresponding to the smallest value of H(p). The second neighbor is chosen as the penguin experiencing the least heat loss among penguins adjacent to the first neighbor and on the huddle’s boundary. Once the mover has been relocated, we may recompute the boundary of the huddle and relocate the effective mover as calculated mathematically as;

 H _(p+1) =Ὠ+ H (p)

Where H _(p+1) denotes the update position of the emperor penguins, Ὠ denotes the heat loss rate and H (p) denotes the largest value of the effective mover.

6. Numerical method of EPO

7. Flowchart of EPO

8. Applications of EPO

• Thermal Environments [11].
• Engineering Design Problems [12].
• High Dimensional data [13].
• No free lunch theorem [14].

9. Advantages of EPO

• EPO analysis the benefits from the body temperature exposed to cold and owing to the warm [15].
• Tested on non-linear and mixed integer structural optimization problems [16].
• Comprehensive investigation of the emperor penguins lifestyle and the extraction of mathematical and physical relations have been analyzed [17].
• Using the spiral like movement to optimize the temperature without need to determine the boundary [18].
• The Performance of EPO algorithm has been evaluated on forty- four linear and non-linear benchmark functions [19].
• High efficiency.

Reference

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