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Advances in Engineering Software 114 (2017) 48–70

Contents lists available at ScienceDirect

Advances in Engineering Software

journal homepage: www.elsevier.com/locate/advengsoft

Research paper

Spotted hyena optimizer: A novel bio-inspired based metaheuristic

technique for engineering applications

Gaurav Dhiman, Vijay Kumar

∗

Computer Science and Engineering Department, Thapar University, Patiala, India

a r t i c l e i n f o

Article history:

Received 27 October 2016

Revised 11 March 2017

Accepted 21 May 2017

Available online 27 May 2017

Keywords:

Optimization

Optimization techniques

Metaheuristics

Constrained optimization

Unconstrained optimization

Benchmark test functions

a b s t r a c t

This paper presents a novel metaheuristic algorithm named as Spotted Hyena Optimizer (SHO) inspired

by the behavior of spotted hyenas. The main concept behind this algorithm is the social relationship

between spotted hyenas and their collaborative behavior. The three basic steps of SHO are searching for

prey, encircling, and attacking prey and all three are mathematically modeled and implemented. The pro-

posed algorithm is compared with eight recently developed metaheuristic algorithms on 29 well-known

benchmark test functions. The convergence and computational complexity is also analyzed. The proposed

algorithm is applied to ﬁve real-life constraint and one unconstrained engineering design problems to

demonstrate their applicability. The experimental results reveal that the proposed algorithm performs

better than the other competitive metaheuristic algorithms.

Introduction

In the last few decades, the increase in complexity of real life

problems has given risen the need of better metaheuristic tech-

niques. These have been used for obtaining the optimal possible

solutions for real-life engineering design problems. These become

more popular due to their eﬃciency and complexity as compared

to other existing classical techniques [35] .

Metaheuristics are broadly classiﬁed into three categories such

as evolutionary-based, physical-based, and swarm-based meth-

ods. The ﬁrst technique is generic population-based metaheuris-

tic which is inspired from biological evolution such as reproduc-

tion, mutation, recombination, and selection. The evolutionary al-

gorithms are inspired by theory of natural selection in which a

population (i.e., a set of solutions) tries to survive based on the

ﬁtness evaluation in a given environment (deﬁned as ﬁtness eval-

uation). Evolutionary algorithms often perform well near optimal

solutions to all types of problems because these methods ideally

do not make any assumption about the basic ﬁtness or adaptive

landscape. Some of the popular evolutionary-based techniques are

Genetic Algorithms (GA) [7] , Genetic Programming (GP) [34] , Evo-

lution Strategy (ES) [6] , and Biogeography-Based Optimizer (BBO)

[56] .

∗

Corresponding author.

E-mail addresses: gdhiman0 0 01@gmail.com (G. Dhiman), vijaykumarchahar

@gmail.com (V. Kumar).

Some of well-known techniques such as Genetic Algorithms(GA)

[7] , Ant Colony Optimization (ACO) [12] , Particle Swarm Optimiza-

tion (PSO) [32] and Differential Evolution (DE) [57] are popular

among different ﬁelds. Due to easy implementation, metaheuris-

tic optimization algorithms are more popular in engineering ap-

plications [4,9,50] ( Fig. 1 ). The second category is physical-based

algorithms. In these algorithms, search agents communicate and

move throughout the search space according to physics rules such

as gravitational force, electromagnetic force, inertia force, and so

on. The name of few algorithms are Simulated Annealing (SA) [33] ,

Gravitational Search Algorithm (GSA) [52] , Big-Bang Big-Crunch

(BBBC) [14] , Charged System Search (CSS) [31] , Black Hole (BH)

[23] algorithm, Central Force Optimization (CFO) [16] , Small-World

Optimization Algorithm (SWOA) [13] , Artiﬁcial Chemical Reaction

Optimization Algorithm (ACROA) [1] , Ray Optimization (RO) algo-

rithm [29] , Galaxy-based Search Algorithm (GbSA) [54] , and Curved

Space Optimization (CSO) [45] .

The last one is swarm-based algorithms which are based on

the collective behavior of social creatures. The collective intelli-

gence is inspired by the interaction of swarm with each other

and their environment. The well-known algorithm of SI technique

is Particle Swarm Optimization (PSO). Another popular swarm-

intelligence technique is Ant Colony Optimization [12] , Monkey

Search [47] , Wolf pack search algorithm [61] , Bee Collecting Pollen

Algorithm (BCPA) [39] , Cuckoo Search (CS) [64] , Dolphin Partner

Optimization (DPO) [55] , Bat-inspired Algorithm (BA) [63] , Fireﬂy

Algorithm (FA) [62] , Hunting Search (HUS) [48] . Generally, swarm-

based algorithms are easier to implement than evolutionary-based

http://dx.doi.org/10.1016/j.advengsoft.2017.05.014

G. Dhiman, V. Kumar / Advances in Engineering Software 114 (2017) 48–70 49

Fig. 1. Classiﬁcation of metaheuristic algorithms.

algorithms due to include fewer operators (i.e., selection, crossover,

mutation). Apart from these, there are other metaheuristic tech-

niques inspired by human behaviors. Some of the popular algo-

rithms are Harmony Search (HS) [19] , Parameter Adaptive Har-

mony Search (PAHS) [35] , Variance-Based Harmony Search [36] ,

Harmony Search-Based Remodularization Algorithm (HSBRA) [3] ,

Tabu (Taboo) Search (TS) [15,21,22] , Group Search Optimizer (GSO)

[24,25] , Imperialist Competitive Algorithm (ICA) [5] , League Cham-

pionship Algorithm (LCA) [28] , Firework Algorithm [59] , Collid-

ing Bodies Optimization (CBO) [30] , Interior Search Algorithm

(ISA) [17] , Mine Blast Algorithm (MBA) [53] , Soccer League Compe-

tition (SLC) algorithm [46] , Seeker Optimization Algorithm (SOA)

[10] , Social-Based Algorithm (SBA) [51] , and Exchange Market Al-

gorithm (EMA) [20] .

The main components of metaheuristic algorithms are explo-

ration and exploitation [2,49] . Exploration ensures the algorithm

to reach different promising regions of the search space, whereas

exploitation ensures the searching of optimal solutions within the

given region [38] . The ﬁne tuning of these components is required

to achieve the optimal solution for a given problem. It is diﬃ-

cult to balance between these components due to stochastic na-

ture of optimization problem. This fact motivates us to develop a

novel metaheuristic algorithm for solving real-life engineering de-

sign problem.

The performance of one optimizer to solve the set of problem

does not guarantee to solve all optimization problems with differ-

ent nature [60] . It is also the motivation of our work and describes

a new metaheuristic based algorithm.

This paper introduces a novel metaheuristic algorithm for op-

timizing constraint and unconstrained design problems. The main

objective of this paper is to develop a novel metaheuristic algo-

rithm named as Spotted hyena Optimization (SHO), which is in-

spired by social hierarchy and hunting behavior of spotted hyenas.

Cohesive clusters can help for eﬃcient co-operation between spot-

ted hyenas. The main steps of SHO are inspired by hunting behav-

ior of spotted hyenas. The performance of the SHO algorithm is

evaluated on twenty-nine benchmark test functions and six real

structural optimization problems. The results demonstrate that the

performance of SHO performs better than the other competitive al-

gorithms.

The rest of this paper is structured as follows:

Section 2 presents the concepts of the proposed SHO algo-

rithm. The experimental results and discussion is presented in

Section 3 . In Section 4 , the performance of SHO is tested on

ﬁve constrained and one unconstrained engineering design prob-

lems and compared with other well-known algorithms. Finally,

the conclusion and some future research directions are given in

Section 5 .

2. Spotted hyena optimizer (SHO)

In this section, the mathematical modeling of proposed algo-

rithm is described in detail.

2.1. Inspiration

Social relationships are dynamic in nature. These are affected

by the changes in the relationships among comprising the net-

work and individuals leaving or joining the population. The social

network analysis of animal behavior has been classiﬁed into three

categories [26] :

•

The ﬁrst category includes environmental factors, such as re-

source availability and competition with other animal species.

•

The second category focuses on social preferences based on in-

dividual behavior or quality.

•

The third category has less attention from scientists which in-

cludes the social relations of species itself.

The social relation between the animals is the inspiration of our

work and correlates this behavior to spotted hyena which is scien-

tiﬁcally named as Crocuta.

Hyenas are large dog-like carnivores. They live in savannas,

grasslands, sub-deserts and forests of both Africa and Asia. They

live 10–12 years in the wild and up to 25 years in imprisonment.

There are four known species of hyena these are, spotted hyena,

50 G. Dhiman, V. Kumar / Advances in Engineering Software 114 (2017) 48–70

Fig. 2. Hunting behavior of spotted hyenas: (A) searching and tracking prey (B) chasing (C) troublesome and encircling (D) immobile situation and attack prey.

striped hyena, brown hyena and aardwolf that differ in size, be-

havior, and type of diet. All these species have a bear-like attitude

as the front legs are longer than the back legs.

Spotted hyenas are skillful hunters and largest of three other

hyena species (i.e., striped, brown, and aardwolf). The Spotted

Hyena is also called Laughing Hyena because its sounds is much

similar to a human laugh. They are so called because there are

spots on their fur which is reddish brown in color with black spots.

Spotted hyenas are complicated, intelligent, and highly social ani-

mals with really dreadful reputation. They have an ability to ﬁght

endlessly over territory and food.

In spotted hyenas family, female members are dominant and

live in their clan. However, male members leave their clan when

they are adults to search and join a new clan. In this new family,

they are lowest ranking members to get their share of the meal. A

male member who has joined the clan always stays with the same

members (friends) for a long time. Whereas a female, is always as-

sured of a stable place. An interesting fact about spotted hyenas

is that they produce sound alert which is very similar to human

laugh to communicate with each other when a new food source is

found.

According to Ilany et al. [26] spotted hyenas usually live and

hunt in groups, rely on a network of trusted friends have more

than 100 members. And to increase their network, they usually tie

up with another spotted hyena that is a friend of a friend or linked

in some way through kinship rather than any unknown spotted

hyena. Spotted hyenas are social animals that can communicate

with each other through specialized calls such as postures and

signals. They use multiple sensory procedures to recognize their

kin and other individuals. They can also recognize third party kin

and rank the relationships between their clan mates and use this

knowledge during social decision making. The spotted hyena track

prey by sight, hearing, and smell. Fig. 2 shows the tracking, chas-

ing, encircling, and attacking mechanism of spotted hyenas. Cohe-

sive clusters are helpful for eﬃcient co-operation between spotted

hyenas and also maximize the ﬁtness. In this work, the hunting

technique and the social relation of spotted hyenas are mathemat-

ically modeled to design SHO and perform optimization.

2.2. Mathematical model and optimization algorithm

In this subsection, the mathematical models of the searching,

encircling, hunting, and attacking prey are provided. Then the SHO

algorithm is outlined.

2.2.1. Encircling prey

Spotted hyenas can be familiar with the location of prey and

encircle them. To mathematically model the social hierarchy of

spotted hyenas, we consider the current best candidate solution is

the target prey or objective which is close to the optimum because

of search space not known a priori. The other search agents will

try to update their positions, after the best search candidate so-

lution is deﬁned, about the best optimal candidate solution. The

Fig. 3. 2D position vectors of spotted hyena.

mathematical model of this behavior is represented by the follow-

ing equations:



= |



B ·



(x ) −



P (x ) |

(1)



P (x + 1) =



(x ) −



E ·



(2)

where



deﬁne the distance between the prey and spotted hyena,

x indicates the current iteration,



B and



E are co-eﬃcient vectors,



indicates the position vector of prey,



P is the position vector of

spotted hyena. However, || and · is the absolute value and multi-

plication with vectors respectively.

The vectors



B and



E are calculated as follows:



B = 2 ·



(3)



E = 2



h ·



−



(4)



h = 5 − (Iteration ∗ (5 /Max

Iteration

))

where, Iteration = 1 , 2 , 3 , . . . , Max

Iteration

(5)

For proper balancing the exploration and exploitation,



h is lin-

early decreased from 5 to 0 over the course of maximum num-

ber of iterations ( Max

Iteration

). Further, this mechanism promotes

more exploitation as the iteration value increases. However,



are random vectors in [0, 1]. Fig. 3 shows the effects of Eqs.

(1) and (2) in two-dimensional environment. In this ﬁgure, the

spotted hyena (A,B) can update its position towards the position

of prey (A

∗

) . By adjusting the value of vectors



B and



E , there

are a different number of places which can be reached about

the current position. The probably updated positions of a spot-

ted hyena in the 3D environment are shown in Fig. 4 . By using

Eqs. (1) and (2) , a spotted hyena can update its position randomly

G. Dhiman, V. Kumar / Advances in Engineering Software 114 (2017) 48–70 51

Fig. 4. 3D position vectors and possible next locations of spotted hyena.

around the prey. Therefore, the same concept can further extend

with n-dimensional search space.

2.2.2. Hunting

Spotted hyenas usually live and hunts in groups and relies on

a network of trusted friends and ability to recognize the location

of prey. To deﬁne the behavior of spotted hyenas mathematically,

we suppose that the best search agent, whichever is optimum, has

knowledge the location of prey. The other search agents make a

cluster, trusted friends group, towards the best search agent and

saved the best solutions obtained so far to update their positions.

The following equations are proposed in this mechanism:



= |



B ·



−



(6)



−



E ·



(7)



k +1

+ . . . +



k + N

(8)

where



deﬁnes the position of ﬁrst best spotted hyena,



indi-

cates the position of other spotted hyenas. Here, N indicates the

number of spotted hyenas which is computed as follows:

N = count

nos

(



h +1



h +2

, . . . , (



M ))

(9)

where



M is a random vector in [0.5, 1], nos deﬁnes the number

of solutions and count all candidate solutions, after addition with



M , which are far similar to the best optimal solution in a given

search space, and



is a group or cluster of N number of optimal

solutions.

2.2.3. Attacking prey (exploitation)

In order to mathematically model for attacking the prey, we de-

crease the value of vector



h . The variation in vector



E is also de-

creased to change the value in vector



h which can decrease from 5

to 0 over the course of iterations. Fig. 6 shows that | E | < 1 forces

the group of spotted hyenas to assault towards the prey. The math-

ematical formulation for attacking the prey is as follows:



P (x + 1) =



(10)

where



P (x + 1) save the best solution and updates the positions

of other search agents according to the position of the best search

agent. The SHO algorithm allows its search agents to update their

position and attack towards the prey.

Fig. 5. Searching prey (| E | > 1).

Fig. 6. Attacking prey (| E | < 1).

2.2.4. Search for prey(exploration)

Spotted hyenas mostly search the prey, according to the posi-

tion of the group or cluster of spotted hyenas which reside in vec-

tor



. They move away from each other to search and attack for

prey. Therefore, we use



E with random values which are greater

than 1 or less than −1 to force the search agents to move far away

from the prey. This mechanism allows the SHO algorithm to search

globally. To ﬁnd a suitable prey, Fig. 5 shows that | E | > 1 facil-

itates the spotted hyenas to move away from the prey. Another

constituent of SHO algorithm which makes possible for exploration



B . In Eq. (3) , the



B vector contains random values which provide

the random weights of prey. To show the more random behavior

of SHO algorithm, assume vector



B > 1 precedence than



B < 1 to

demonstrate the effect in the distance as may be seen in Eq. (3) .

This will helpful for exploration and local optima avoidance. De-

pending on the position of a spotted hyenas, it can randomly de-

cide a weight to the prey and possible makes it rigid or beyond to

reach for spotted hyenas. We intentionally need vector



B to provide

random values for exploration not only during initial iterations but

for also ﬁnal iterations. This mechanism is very helpful to avoid lo-

cal optima problem, more than ever in the ﬁnal iterations. Finally,

the SHO algorithm is terminated by satisfying termination criteria.

The pseudo code of the SHO algorithm shows that how SHO can

solve optimization problems, some points may be noted as follows:

•

The proposed algorithm saves the best solutions obtained so far

over the course of the iteration.

•

The proposed encircling mechanism deﬁnes a circle-shaped the

neighborhood around the solutions which can be extended to

higher dimensions as a hyper-sphere.

•

Random vectors



B and



E assist candidate solutions to have

hyper-spheres with different random positions.

•

The proposed hunting method allows candidate solutions to lo-

cate the probable position of the prey.

•

The possibility of exploration and exploitation by the ad-

justed values of the vectors



E and



h and allows SHO to easily

changeover between exploration and exploitation.

•

With vector



E , half of the iterations are dedicated to searching

(exploration) (| E | ≥ 1) and the other half are devoted to hunting

(exploitation) (| E | ≤ 1).

52 G. Dhiman, V. Kumar / Advances in Engineering Software 114 (2017) 48–70

2.3. Steps and ﬂowchart of SHO

The steps of SHO are summarized as follows:

Step 1: Initialize the spotted hyenas population P

where i =

1 , 2 , . . . , n .

Step 2: Choose the initial parameters of SHO: h, B, E , and N and

deﬁne the maximum number of iterations.

Step 3: Calculate the ﬁtness value of each search agent.

Step 4: The best search agent is explored in the given search

space.

Step 5: Deﬁne the group of optimal solutions, i.e cluster using

Eqs. (8) and (9) until the satisfactory result is found.

Step 6: Update the positions of search agents using Eq. (10) .

Step 7: Check whether any search agent goes beyond the

boundary in a given search space and adjust it.

Step 8: Calculate the update search agent ﬁtness value and up-

date the vector P

if there is a better solution than previous

optimal solution.

Step 9: Update the group of spotted hyenas C

to updated

search agent ﬁtness value.

Step 10: If the stopping criterion is satisﬁed, the algorithm will

be stopped. Otherwise, return to Step 5.

Step 11: Return the best optimal solution, after stopping criteria

is satisﬁed, which is obtained so far.

Algorithm 1 Spotted Hyena Optimizer.

Input: the spotted hyenas population P

( i = 1 , 2 , . . . , n )

Output: the best search agent

procedure SHO

Initialize the parameters h, B, E, and N

Calculate the ﬁtness of each search agent

4: P

= the best search agent

= the group or cluster of all far optimal solutions

while ( x < Max number of iterat ions ) do

for each search agent do

Update the position of current

agent by Eq. (10)

end for

10: Update h, B, E, and N

11:

Check if any search agent goes beyond the

given search space and then adjust it

12: Calculate the ﬁtness of each search agent

13:

Update P

if there is a better solution than

previous optimal solution

14: Update the group C

w.r.t P

15:

x = x + 1

16:

end while

17: return P

18:

end procedure

2.4. Computational complexity

In this subsection, the computational complexity of proposed

algorithm is discussed. Both the time and space complexities of

proposed algorithm is given below ( Fig. 7 ).

2.4.1. Time complexity

1. Initialization of SHO population needs O ( n × dim ) time where

n indicates the number of iterations to generate random pop-

ulation which is based on the number of search agents, lower-

bound, and upperbound of a test function. However, dim indi-

cates the dimension of a test function to check and adjust the

solutions which are goes beyond the search space.

Fig. 7. Flowchart of the proposed SHO.

2. In the next step, the ﬁtness of each agent requires O ( Max

iter

× dim ) time where Max

iter

is the maximum number of itera-

tions to simulate the proposed algorithm.

3. It requires O ( Max

iter

× N ) time to deﬁne the group of spotted

hyenas where Max

iter

is the maximum iteration of an algorithm

and N is the counting value of spotted hyenas.

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