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1

INFO: An Efficient

Optimization Algorithm based

on Weighted Mean of Vectors

Iman Ahmadianfara*, Ali Asghar Heidarib,c, Saeed Noshadiana,

Huiling Chend**, Amir H Gandomie

a Department of Civil Engineering, Behbahan Khatam Alanbia University of Technology,

Behbahan, Iran

Email: im.ahmadian@gmai.com, i.ahmadianfar@bkatu.ac.ir (Iman Ahmadianfar).

Saeed.noshadian@gmail.com (Saeed Noshadian).

b School of Surveying and Geospatial Engineering, College of Engineering, University of

Tehran, Tehran 1439957131, Iran

Email: as_heidari@ut.ac.ir, aliasghar68@gmail.com

c Department of Computer Science, School of Computing, National University of Singapore,

Singapore 117417, Singapore

Email: aliasgha@comp.nus.edu.sg, t0917038@u.nus.edu

d College of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou,

Zhejiang 325035, China

Email: chenhuiling.jlu@gmail.com

e Faculty of Engineering & Information Technology, University of Technology Sydney, NSW

2007, Australia

Email: gandomi@uts.edu.au

Corresponding Authors: Iman Ahmadianfar and Huiling Chen

E-mail: im.ahmadian@gmail.com (Iman Ahmadianfar), chenhuiling.jlu@gmail.com (Huiling

Chen)

All codes and source files are available at https://aliasgharheidari.com/INFO.html

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Abstract

This study presents the analysis and principle of an innovative optimizer named weIghted

meaN oF vectOrs (INFO) to optimize different problems. INFO is a modified weight mean

method, whereby the weighted mean idea is employed for a solid structure and updating the

vectors’ position using three core procedures: updating rule, vector combining, and a local search.

The updating rule stage is based on a mean-based law and convergence acceleration to generate

new vectors. The vector combining stage creates a combination of obtained vectors with the

updating rule to achieve a promising solution. The updating rule and vector combining steps

were improved in INFO to increase the exploration and exploitation capacities. Moreover,

the local search stage helps this algorithm escape low-accuracy solutions and improve

exploitation and convergence. The performance of INFO was evaluated in 48 mathematical

test functions, and five constrained engineering test cases. According to the literature, the

results demonstrate that INFO outperforms other basic and advanced methods in terms of

exploration and exploitation. In the case of engineering problems, the results indicate that the

INFO can converged to 0.99% of the global optimum solution. Hence, the INFO algorithm

is a promising tool for optimal designs in optimization problems, which stems from the

considerable efficiency of this algorithm for optimizing constrained cases.

The source codes of this algorithm will be publicly available at https://imanahmadianfar.com.

and https://aliasgharheidari.com/INFO.html.

Keywords: Optimization; Swam-intelligence; Exploration; Exploitation; Weighted Mean of

Vectors Algorithm

1. Introduction

With the development of society, people will face more and more complex problems.

However, solving a class of complex problems is the essential requirement for promoting

social development. Although many traditional numerical and analytical methods have carried

out relevant analysis research, some deterministic methods cannot provide a fitting solution

to solve several challenging problems with non-convex and highly non-linear search domains

since the complexity and dimensions of these problems grow exponentially. Optimizing the

problems by applying some deterministic methods, such as the Lagrange and Simplex

methods, requires some initial information of the problem and complicated computations.

Thus, exploring global optimum solution problems using such methods for those levels of

problems is not always possible or feasible [1]. Therefore, it is still urgent to develop an

efficient method to solve the increasingly complex optimization problems. Actually,

optimization methods can have multiple forms and formulations, maybe no limit in form, and

what they essential for them in stochastic class is a core for exploration and a core for

exploitation, which can be utilized to deal with those forms of problems, such as multi-

objective optimization, fuzzy optimization, robust optimization, memetic optimization, large

scale optimization, many-objective optimization methods, and single-objective optimization.

One common optimization method, named swarm intelligence (SI) algorithms, is swarm-

based optimization based on the evolution of an initial set of agents and attraction of agents

towards better solutions, which, in an extreme case, is the optimum solution and avoids locally

optimal solutions. The swarm intelligence optimization algorithm has intelligent

characteristics such as self-adaptation, self-learning, and self-organization and is convenient

for large-scale parallel computing. It is a trendy optimization technology.

In recent years, some classes of swarm-based optimization algorithms have been

applied as simple and reliable methods for realizing the solutions of problems in both the

computer science field and industry. Numerous researchers have demonstrated that swarm-

All codes and source files are available at https://aliasgharheidari.com/INFO.html

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based optimization is very promising for tackling many challenging problems [2-4]. Some

algorithms employ methods that mimic natural evolutionary mechanisms and basic genetic

rules, such as selection, reproduction, mutation, and migration [5]. One of the most popular

evolutionary methods is the genetic algorithm (GA) introduced by Holland [6]. With its unique

three core operations of crossover, variation, and selection, GA has achieved outstanding

performance in many optimization problems, such as twice continuously differentiable NLP

problems [7], predicting production, and neural architectures searching. Other well-regarded

evolutionary algorithms include differential evolution (DE) [8] and evolutionary strategies

(ES) [9]. This kind of evolutionary algorithm simulates the way of biological evolution in

nature and has strong adaptability to problems. Moreover, the rise of deep neural networks

(DNN) in recent years has made people pay more attention to how to design neural network

architecture automatically. Therefore, network architecture search (NAS) based on

evolutionary algorithms has become a hot topic [10]. Some methods are motivated by physical

laws, such as simulated annealing (SA) [11]. As one of the most well-known methods in this

family presented by Kirkpatrick et al. [11], SA simulates the annealing mechanism utilized in

physical material sciences. Also, with its excellent local search capabilities, SA can find more

potential solutions in many engineering problems than other traditional SI algorithms [12-14].

One of the latest well-established methods is the gradient-based optimizer (GBO)

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, which

considers Newtonian logic to explore suitable regions and achieve the global solution [15].

The method has been applied in many fields, including feature selection[16] and parameter

estimation of photovoltaic models[16]. Most swarm methods mimic the equations of particle

swarm optimization (PSO) by varying the basis of inspiration around collective social

behaviors of animal groups [17]. Particle swarm optimization (PSO) is one of the most

successful algorithms in this class, which was inspired by birds' social and individual

intelligence when flocking [18]. In detail, PSO has a few parameters that need to be adjusted,

also, unlike other methods, PSO has a memory machine, and the knowledge of particles with

better performance can be preserved, which can help the algorithm find the optimal solution

more quickly. Currently, PSO has taken its place in the fields of large-scale optimisation

problems[19], feature selection[20], single-objective optimization problem[21], multi-

objective optimisation problems[22], and high-dimensional expensive problem[23]. Ant

colony optimization (ACO) is another popular approach based on ants' foraging behavior

[24]. In particular, the concept of pheromone is a major feature of ACO. According to

pheromone secreted by the ants in the process of searching for food, it can help the population

to find a better solution at a faster rate. As soon as ACO was proposed, it was applied to the

traveling salesman problem of 3-spots [25] and some complex optimization problems[26], and

achieved satisfactory results.

While these optimization methods can solve various challenging and real optimization

problems, the No Free Lunch (NFL) theorem authorizes researchers to present a new variant

of methods or a new optimizer from scratch [27]. This theory states that no optimization

method can work as the best tool for all problems. Accordingly, one algorithm can be the

most suitable approach to solve several problems but is incompetent for other optimization

problems. Hence, it can be declared that this theory is the basis of many studies in this field.

In this research, we were motivated to improve upon novel metaheuristic methods that suffer

from weak performance, have verification bias, and underperform compared to other existing

methods [28-30]. As such, the proposed INFO algorithm is a forward-thinking, innovative

attempt against such methods that provides a promising platform for the future of

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https://imanahmadianfar.com

All codes and source files are available at https://aliasgharheidari.com/INFO.html

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optimization literature in computer science. Furthermore, we aim to apply this method to a

variety of optimization problems and make it a scalable optimizer.

In this paper, we designed a new optimizer (INFO) by modifying the weighted mean

method and updating the vectors' position, which can help form a more robust structure. In

detail, updating rule, vector combining, and local search are the three core processes of INFO.

Unlike other methods, the updating rule based on the mean is used to generate new vectors

in INFO, thus accelerating the convergence speed. In the vector combination stage, two

vectors acquired in the vector update stage are combined to produce a new vector for

improving local search ability. This operation ensures the diversity of the population to a

certain extent. Taking into account the global optimal position and the mean-based rule, a

local operation is executed, which can effectively improve the problem of INPO being

vulnerable to local optimal. This work's primary goal was to introduce the above three core

processions for optimizing various kinds of optimization cases and engineering problems,

such as structural and mechanical engineering problems and water resources systems. The

INFO algorithm employs the concept of weighted mean to move agents toward a better

position. This main motive behind INFO emphasizes its performance aspects to potentially

solve some of the optimization problems that other methods cannot solve. It should be noted

that there is no inspiration part in INFO, and it is tried to move the field to go beyond the

metaphor.

The rest of this paper is organized as follows. In Sections 2 and 3, the main structures

of INFO are described in detail. The set of mathematical benchmark functions employed to

assess the efficiency of INFO is presented in Section 4. Section 5 solves four real engineering

problems to show the capability of the proposed algorithm. Lastly, Section 6 expresses the

conclusions of this study and gives some ideas for future researches.

2. Literature review

This section describes the previous studies on optimization methods and presents this

research's primary motivation. Generally, evolutionary algorithms are classified into two types:

single-based and population-based algorithms [17, 31]. In the first case, the algorithm's search

process begins with a single solution and updates its position during the optimization process.

The most well-known single-solution-based algorithms include simulated annealing (SA) [11],

tabu search (TS) [32], and hill-climbing [33]. These algorithms allow easy implementation and

require only a small number of function evaluations during optimization. However, the

disadvantages are the high possibility of trapping in local optima and failure to exchange

information because these methods have only a single trend.

Conversely, the optimization process in population-based algorithms begins with a

set of solutions and updates their position during optimization. GA, DE, PSO, artificial bee

colony (ABC) [34], ant colony optimization (ACO) [35-37], slime mould algorithm (SMA)

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[38], and Harris hawks optimization (HHO)

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[39-41] are some of the population-based

algorithms. These methods have a high capacity to escape local optimal solutions because they

use a set of solutions during optimization. Moreover, the exchange of information can be

shared between solutions, which helps them to better search in difficult search spaces.

However, these algorithms require a large number of function evaluations during optimization

and high computational costs.

According to the above discussion, the population-based algorithms are considered

more reliable and robust optimization methods than single-solution-based algorithms.

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https://aliasgharheidari.com/SMA.html

3

https://aliasgharheidari.com/HHO.html

All codes and source files are available at https://aliasgharheidari.com/INFO.html

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Generally, an algorithm's best formulation is explored by evaluating it on different types of

benchmark and engineering problems.

Ordinarily, optimizers employ one or more operators to perform two phases:

exploration and exploitation. An optimization algorithm requires a search mechanism to find

promising areas in the search space, which is done in the exploration phase. The exploitation

phase improves the local search ability and convergence speed to achieve promising areas.

The balance between these two phases is a challenging issue for any optimization algorithm.

According to previous studies, no precise rule has been established to distinguish the most

appropriate time to transit from exploration to exploitation due to the unexplored form of

search spaces and the random nature of this type of optimizer [17, 31]. Therefore, realizing

this issue is essential to design a robust and reliable optimization algorithm.

Considering the main challenges of creating a high-performance optimization

algorithm and all critical points highlighted in the literature above [42-44], we introduce an

efficient optimizer based on the concept of the weighted mean of vectors. By avoiding a basis

of nature inspiration, INFO offers a promising method to avoid and reduce the challenges of

other optimization algorithms, thus providing a strong step in the direction towards a

metaphor-free class of optimization algorithms.

3. Definition of weighted mean

The optimization algorithm introduced in this study is based on a weighted mean,

which demonstrates a unique location in an object or system [45]. A detailed definition of this

concept is subsequently provided.

3.1. Mathematical definition of weighted mean

The mean of a set of vectors is described as the average of their positions (xi), as

weighted by the fitness of a vector (wi) [45]. In fact, this concept is used due to its simplicity

and ease of implementation. Fig. 1 depicts the weighted mean of the set of solutions (vectors),

in which the solutions with bigger weights are more effective in calculating the weighted mean

of solutions.

The formulation of weighted mean (WM) is defined by Eq. (1) [45]:

1

1

N

ii

iN

i

i

xw

WM w

=

=

=

(1)

where N is the number of vectors.

To provide a better explanation, WM can be considered as two vectors, as shown in

Eq. (1.1) [45]:

1 1 2 2

12

x w x w

WM ww

+

=+

(1.1)

In this study, each vector's weight was calculated based on a wavelet function (WF)

[46, 47]. Generally, the wavelet is a useful tool for modeling seismic signals by compounding

translations and dilations of an oscillatory function (i.e., mother wavelet) with a finite period.

This function is employed to create effective fluctuations during the optimization process.

Fig. 2 displays the mother wavelet used in this study, which is defined as:

2

cos( ) exp( )

x

wx

= −

(2)

All codes and source files are available at https://aliasgharheidari.com/INFO.html

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where

is a constant number called the dilation parameter.

Fig. 1. The weighted mean of a set of solutions

Fig. 2. Mother wavelet

Figs. 3a and 3b display three vectors, and the differential between them are shown in

Fig 3c. The weighted mean of vectors is calculated by Eq. (3):

1 1 2 2 1 3 3 2 3

1 2 3

( ) ( ) ( )w x x w x x w x x

WM w w w

− + − + −

=++

(3)

in which

12

1 1 2 ( ) ( )

cos(( ( ) ( )) ) exp( )

f x f x

w f x f x

−

= − +

(3.1)

13

2 1 3 ( ) ( )

cos(( ( ) ( )) ) exp( )

f x f x

w f x f x

−

= − +

(3.2)

23

3 2 3 ( ) ( )

cos(( ( ) ( )) ) exp( )

f x f x

w f x f x

−

= − +

(3.3)

All codes and source files are available at https://aliasgharheidari.com/INFO.html

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where

()fx

denotes the fitness function of the vector

x

.

Fig. 3. The weighted mean of vectors for three vectors

4. Weighted mean of vectors (INFO) algorithm

The weighted mean of vectors algorithm (INFO) is a population-based optimization

algorithm that calculates the weighted mean for a set of vectors in the search space. In the

proposed algorithm, the population is comprised of a set of vectors that demonstrate possible

solutions. The INFO algorithm finds the optimal solution over several successive generations.

The three operators update the vectors' positions in each generation:

• Stage 1: Updating rule

• Stage 2: Vector combining

• Stage 3: Local search

Herein, the problem of minimizing the objective function is considered as an example.

4.1. Initialization stage

The INFO algorithm is comprised of a population of Np vector in D dimensional

search domain (

NplxxxX gDl

g

l

g

l

gjl ...,,2,1},...,,,{ ,2,1,, ==

). In this step, some control parameters

are introduced and defined for the INFO algorithm. There are two main parameters: weighted

mean factor and scaling factor .

Generally speaking, the scaling rate is used to amplify the obtained vector via the

updating rule operator, which is dependent on the size of the search domain. The

factor is

used to scale the weighted mean of vectors. Its value is specified based on the feasible search

space of problems and reduced according to an exponential formula. These two parameters do

not need to be adjusted by the user and change dynamically based on generation. The INFO

algorithm uses a simple method to generate the initial vectors called random generation.

4.2. Updating rule stage

In the INFO algorithm, the updating rule operator increases the population's diversity

during the search procedure. This operator uses the weighted mean of vectors in order to

All codes and source files are available at https://aliasgharheidari.com/INFO.html

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create new vectors. Indeed, this operator distinguishes the INFO algorithm from other

algorithms and consists of two main parts. In the first part, a mean-based rule is extracted

from the weighted mean for a set of random vectors. The mean-based method begins from a

random initial solution and moves to the next solution using the weighted mean information

of a set of randomly selected vectors. The second part is convergence acceleration, which

enhances convergence speed and promotes the algorithm's performance to reach optimal

solutions.

In general, INFO first employs a set of selected randomly differential vectors to obtain

the weighted mean of vectors rather than move the current vector toward a better solution.

In this work, increasing the population's diversity is considered the MeanRule based on the

best, better, and worst solutions. It should be noted that the better solution is randomly

determined from the top 5 solutions (regarding the objective function value). Therefore, the

mean-based rule is conducted to the MeanRule, as defined in Eq. (4):

1 (1 ) 2

1,2,...,

gg

ll

MeanRule r WM r WM

l Np

= + −

=

(4)

1 1 2 2 1 3 3 2 3

1 2 3

( ) ( ) ( )

1,

1,2,...,

ga a a a a a

lw x x w x x w x x

WM rand

w w w

l Np

− + − + −

= +

+ + +

=

(4.1)

where

12

1 1 2 ( ) ( )

cos(( ( ) ( )) ) exp( )

aa

aa f x f x

w f x f x

−

= − + −

(4.2)

13

2 1 3 ( ) ( )

cos(( ( ) ( )) ) exp( )

aa

aa f x f x

w f x f x

−

= − + −

(4.3)

23

3 2 3 ( ) ( )

cos(( ( ) ( )) ) exp( )

aa

aa f x f x

w f x f x

−

= − + −

(4.4)

1 2 3

max( ( ), ( ), ( ))

a a a

f x f x f x

=

(4.5)

1 2 3

1 2 3

( ) ( ) ( )

2,

1,2,...,

gbs bt bs ws bt ws

l

w x x w x x w x x

WM rand

w w w

l Np

− + − + −

= +

+ + +

=

(4.6)

where

1( ) ( )

cos(( ( ) ( )) ) exp( )

bs bt

bs bt f x f x

w f x f x

−

= − + −

(4.7)

2( ) ( )

cos(( ( ) ( )) ) exp( )

bs ws

bs ws f x f x

w f x f x

−

= − + −

(4.8)

All codes and source files are available at https://aliasgharheidari.com/INFO.html

9

3( ) ( )

cos(( ( ) ( )) ) exp( )

bt ws

bt ws f x f x

w f x f x

−

= − + −

(4.9)

()

ws

fx

=

(4.10)

where

()fx

is the value of the objective function;

1 2 3a a a l

are different integers

randomly selected from the range [1, NP];

is a constant number and has a very small value;

randn is a normally distributed random value;

bs

x

,

bt

x

, and

ws

x

are the best, better, and worst

solutions among all vectors in the population for the gth generation, respectively. In fact, these

solutions are determined after sorting the solution at each iteration. r is a random number

within the range [0, 0.5]; and

1

w

,

2

w

, and

3

w

are three WFs to calculate the weighted mean

of vectors that help the proposed INFO algorithm to search in the solution space globally.

In fact, the WFs are used to vary the MeanRule space according to the wavelet theory,

which is considered for two reasons: (1) to assist the algorithm to explore the search space

more effectively and achieve better solutions by creating efficient oscillation during the

optimization procedure; and (2) to generate fine-tuning by controlling the dilation parameter

introduced in the WFs, which is used to adjust the amplitude of WF. In this study, the value

of the dilation parameter was varied using Eq. (4.10) during the optimization process. In Eq.

(5),

is the scale factor, and

can be changed based on an exponential function defined in

(5.1):

2rand

= −

(5)

2exp( 4 )

g

Maxg

= −

(5.1)

where Maxg is the maximum number of generations.

The convergence acceleration part (CA) is also added to the updating rule operator to

promote global search ability, using the best vector to move the current vector in a search

space. In the INFO algorithm, it is supposed that the best solution is the nearest solution to

global optima. In fact, CA helps vectors move in a better direction. The CA presented in Eq.

(6) is multiplied by a random number in the range [0,1] (

rand

) to ensure that each vector has

a different step size in each generation in INFO:

1

1

()

( ( ) ( ) )

bs a

bs a

xx

CA randn f x f x

−

= −+

(6)

where

randn

is a random number with a normal distribution.

Finally, the new vector is calculated using Equation (7):

gg

ll

z x MeanRule CA

= + +

(7)

An optimization algorithm should generally search globally to discover the search

domain's promising spaces (exploration phase). Accordingly, the proposed updating rule

based on

bs

x

,

bt

x

,

g

l

x

and

1

g

a

x

is defined using the following scheme:

All codes and source files are available at https://aliasgharheidari.com/INFO.html

10

(8)

where

g

l

z1

and

g

l

z2

are the new vectors in the gth generation; and

is the scaling rate of a vector,

as defined in Eq. (9). It should be noted that in Eq. (9),

can be changed based on an

exponential function defined in Eq. (9.1):

−= rand2

(9)

exp( )

g

cd

Maxg

= −

(9.1)

where c and d are constant numbers equal to 2 and 4, respectively. It is worth noting that for

large values of the parameter

, the current position tends to diverge from the weighted mean

of vectors (exploration search), while small values of this parameter force the current position

to move toward the weighted mean of vectors (exploitation search).

4.3. Vector combining stage

In this study, for enhancing the population’s diversity in INFO, the two vectors

calculated in the previous section (

g

l

z1

and

g

l

z2

) are combined with vector

g

l

x

regarding the

condition to generate the new vector

g

l

u

, according to Eq. (10). In fact, this

operator is used to promote the local search ability to provide a new and promising vector:

(10.1)

(10.2)

(10.3)

whereis the obtained vector using the vector combining in gth generation; and

is equal

to

0.05 randn

.

4.4. Local search stage

All codes and source files are available at https://aliasgharheidari.com/INFO.html

11

Effective local search ability can prevent INFO from deception and dropping into

locally optimal solutions. The local operator is considered using the global position (

g

best

x

) and

the mean-based rule defined in Eq. (11) to further promote the exploitation, search, and

convergence to global optima. According to this operator, a novel vector can be produced

around

g

best

x

, if r < 0.5, where rand is a random value in [0, 1]:

1

12

0.5

0.5

( ( ))

( ( ))

g g g

l bs bs a

g

l rnd bs rnd

if rand

if rand

u x randn MeanRule randn x x

else

u x randn MeanRule randn x x

end

end

= + + −

= + + −

(11.1)

(11.2)

in which

(1 ) ( (1 ) )

rnd avg bt bs

x x x x

= + − + −

(11.3)

3

()

3

ab

avg x x x

x++

=

(11.4)

where

denotes a random number in the range of (0, 1); and

rnd

x

is a new solution, which

combines the components of the solutions,

avg

x

,

bt

x

, and

bs

x

, randomly. This increases the

randomness nature of the proposed algorithm to better search in the solution space.

1

and

2

are two random numbers defined as:

(11.5)

20.5

1

rand if p

otherwise

=

(11.6)

where p denotes a random number in the range of (0, 1). The random numbers

1

and

2

can

increase the impact of the best position on the vector. Finally, the proposed INFO algorithm

is presented in Algorithm 1, and Fig. 4 depicts the flowchart of the proposed algorithm.

The calculation complexity of an optimization algorithm is used to assess the run-

time, which is determined based on the algorithm's structure. INFO’s computational

complexity depends on the number of vectors, the total number of iterations and the number

of objects and is calculated as follows:

( ) ( ( )) ( )O CMV O T N d O TNd= =

(12)

where N is the number of vectors (population size), T is maximum generations, and d is the

number of objects.

All codes and source files are available at https://aliasgharheidari.com/INFO.html

12

To demonstrate the potential of INFO to solve optimization problems, its capabilities

are described below:

• INFO generates and promotes a set of random vectors for a problem and inherently

has a high ability to explore and escape local optimal solutions to single-solution-based

algorithms.

• The proposed updating rule in the INFO mechanism uses the mean rule and

convergence acceleration part to find the search space's up-and-coming areas.

• The proposed vector combining operator can explore the search space to improve the

search capability and local optima avoidance.

• Adaptive parameters smoothly implement the transition from exploration to

exploitation.

• A complement strategy called a local search operator is used to promote the exploitation

and convergence speed further.

Algorithm 1. Pseudo-code of the INFO algorithm.

STEP 1. Initialization

Set parameters Np and Maxg

Produce an initial population

}...,,{ 000 NpiXXP =

Calculate the objective function value of each vector

NpiXf i...,,1),(0=

Determine the best vector

bs

x

STEP 2. for g = 1 to Maxg do

for i = 1 to Np do

Select randomly

icba

within the range [1, Np]

►Updating rule stage

Calculate the vectors

g

i

z1

and

g

i

z2

using Eq. (8)

►Vector combining stage

Calculate the vector

g

i

u

using Eq. (10)

► Local search stage

Calculate the local search operator using Eq. (11)

Calculate the objective function value

)( ,

gji

uf

if

)()( ,, gji

gji xfuf

then

gji

gji ux ,

1

,=

+

Otherwise

gji

gji xx ,

1

,=

+

end for

Update the best vector (

bs

x

(

end for

STEP 3. Return Vector

gjbest

x,

as the final solution

All codes and source files are available at https://aliasgharheidari.com/INFO.html

13

• A variable is used as the global best position to record an appropriate approximation

of the global optimum and promote it during optimization.

• Since the vectors can change their positions according to the best position generated

so far, this will tend toward the best regions of the search spaces during the

optimization.

The next sections verify the performance of INFO in several test functions and real

engineering problems.

Fig. 4. Flowchart of INFO

5. Results and discussion

To evaluate and confirm the efficiency of an optimization algorithm, several test

problems should be considered. Therefore, this work tested the INFO algorithm's

performance on 19 mathematical benchmark functions, 13 of which (f1-f7 and f8-f13) have been

widely utilized in previous studies [38, 39, 48] and have unimodal (UF) and multimodal (MF)

search spaces, respectively. Functions f14- f19 are composite functions that have also been

considered in several previous studies [17, 31]. In the challenging composite functions (CFs),

the global solution position is shifted to a random position, and the functions are rotated,

which occasionally places the global solution within infeasible space boundaries occasionally

and combines variants of the benchmark functions.

A detailed explanation of these functions is reported in Tables 2-4. INFO was

compared with the GWO, GSA, SCA, GA, PSO, and BA optimization algorithms. We

followed fair comparison standards. For all the optimizers, the population size and the total

number of iterations were set to 30 and 500. The values of the main parameters for all

algorithms are given in Table 1. It is pertinent to mention that all of the control parameters

were set based on their developer or within the range of the suggestions to achieve the best

All codes and source files are available at https://aliasgharheidari.com/INFO.html

14

efficiency of the optimizers. The parameter settings of GWO and SCA were obtained from

previous works [49]. The benchmark functions for each optimization algorithm were tested

30 times. Table 5 presents the average and standard deviation of the fitness functions for the

30 runs.

Table 1. Values of control parameters for all comparative algorithms

Algorithms

Values of parameters

GWO

Convergence constant a = [2, 0]

BA

A (loudness) = 0.5, r (plus rate) = 0.5,

= 0, = 2

GA

Crossover probability =0.8 , mutation

probability = 0.05

PSO

c1 = c2 = 1.5, w (inertial weight) linearly

decreased from 0.7 to 0.3

GSA

G0 (initial gravitational constant) = 100,

= 20

SCA

A = 2

INFO

= 2, = 4

Table 2. UF test problems

Function

Dimension

Range

30

[-100,100]

0

30

[-100,10]

0

30

[-100,100]

0

,

30

[-100,100]

0

30

[-30,30]

0

30

[-100,100]

0

,

30

[-1.28,1.28]

0

All codes and source files are available at https://aliasgharheidari.com/INFO.html

15

Table 3. MF test problems

Function

Dimension

Range

30

[-500,500]

-418.98295

30

[-5.12,5.12]

0

30

[-32,32]

0

30

[-600,600]

0

,,,

,,,

30

[-50,50]

0

,,,

30

[-50,50]

0

All codes and source files are available at https://aliasgharheidari.com/INFO.html

16

Table 4. CF test problems

Function

Dimension

Range

,,,, Sphere Function

σ1,σ2,σ3,…,σ10=1,1,1,…,1

λ1,λ2,λ3,…,λ10=5100

, 5 100

, 5 100

,…, 5 100

30

[-5,5]

0

,,,, Griewank’s Function

σ1,σ2,σ3,…,σ10=1,1,1,…,1

λ1,λ2,λ3,…,λ10=5100

, 5 100

, 5 100

,…, 5 100

30

[-5,5]

0

,,,, Griewank’s Function

σ1,σ2,σ3,…,σ10=1,1,1,…,1

λ1,λ2,λ3,…,λ10=1,1,1,…,1

30

[-5,5]

0

,Ackley’s Function

, Rasrigin’s Function

, Sphere’s Function

,Weierstras’s Function

, Griewank’s Function

σ1,σ2,σ3,…,σ10=1,2,1.5,1.5,1,1,1.5,1.5,2,2

λ1,λ2,λ3,…,λ10=2*5 32

, 5 32

,2*1,1, 2*5 100, 5100,

2*10,10, 2*5 60, 560,

30

[-5,5]

0

, Rasrigin’s Function

, Weierstras’s Function

, Griewank’s Function

,Ackley’s Function

, Sphere’s Function

σ1,σ2,σ3,…,σ10=1,1,1,…,1

λ1,λ2,λ3,…,λ10=1,1,10,10, 5 60, 560,

532, 532,

5100,

5100

30

[-5,5]

0

, Rasrigin’s Function

, Weierstras’s Function

, Griewank’s Function

,Ackley’s Function

, Sphere’s Function

σ1,σ2,σ3,…,σ10=1,1,1,…,1

λ1,λ2,λ3,…,λ10=1,1,10,10, 5 60, 560,

532, 532,

5100,

5100

30

[-5,5]

0

All codes and source files are available at https://aliasgharheidari.com/INFO.html

17

Table 5. Statistical results and comparison for test functions

Function

INFO

GWO

GSA

SCA

PSO

BA

GA

f1

Mean

2.59E-43

2.02E-27

2.49E-01

1.49E+00

2.43E-16

3.92E+00

1.74E-01

SD

1.04E-43

4.27E-27

2.19E-01

8.94E-01

1.09E-16

6.08E+00

2.66E-02

f2

Mean

3.23E-21

8.03E-17

8.36E-01

3.30E-01

1.26E-07

1.37E-02

1.68E-01

SD

2.29E-21

5.53E-17

2.56E-01

8.34E-02

1.90E-07

2.62E-02

6.61E-02

f3

Mean

6.46E-39

1.72E-05

1.22E+02

1.55E+03

8.55E+02

9.21E+03

3.50E-01

SD

2.98E-38

6.95E-05

4.21E+01

1.01E+03

2.15E+02

5.30E+03

1.76E-01

f4

Mean

8.28E-22

8.51E-07

1.51E+00

1.02E+01

7.10E+00

3.71E+01

3.57E-01

SD

4.49E-22

1.33E-06

2.22E-01

2.71E+00

2.54E+00

1.13E+01

2.81E-01

f5

Mean

2.47E+01

2.72E+01

2.62E+02

2.78E+02

4.01E+01

1.74E+04

2.78E+01

SD

7.45E-01

8.02E-01

1.51E+02

1.45E+02

2.06E+01

2.83E+04

2.33E+00

f6

Mean

1.54E-06

6.75E-01

2.53E-01

1.89E+00

3.73E+00

2.19E+01

1.60E-02

SD

3.93E-06

3.27E-01

1.79E-01

7.01E-01

4.08E+00

3.03E+01

8.51E-02

f7

Mean

1.62E-03

1.79E-03

1.56E+00

1.15E-01

8.73E-02

1.43E-01

8.55E-02

SD

1.34E-03

6.61E-04

1.12E+00

3.86E-02

3.45E-02

2.29E-01

1.95E-02

f8

Mean

-9.47E+03

-6.16E+03

-6.23E+03

-6.63E+03

-2.76E+03

-3.70E+03

-5.61E+03

SD

6.40E+02

8.55E+02

1.07E+03

6.69E+02

5.19E+02

2.73E+02

6.76E+02

f9

Mean

0.00E+00

3.02E+00

1.02E+02

9.27E+00

2.49E+01

3.60E+01

3.26E+00

SD

0.00E+00

4.52E+00

2.24E+01

3.17E+00

4.67E+00

4.00E+01

4.51E+00

f10

Mean

8.88E-16

1.03E-13

1.01E+00

9.59E-01

1.11E-08

1.65E+01

5.70E+00

SD

0.00E+00

1.82E-14

5.90E-01

5.92E-01

2.64E-09

7.10E+00

3.10E+00

f11

Mean

0.00E+00

2.97E-03

2.06E-02

9.71E-01

2.78E+01

1.03E+00

3.91E-01

SD

0.00E+00

6.45E-03

8.48E-03

7.43E-02

7.07E+00

3.91E-01

6.41E-01

f12

Mean

1.04E-02

5.93E-02

1.96E-02

1.64E+00

2.28E+00

8.65E+00

6.98E-01

SD

3.16E-02

9.16E-02

3.96E-02

1.63E+00

1.05E+00

7.77E+00

9.24E-01

f13

Mean

4.30E-02

6.59E-01

7.31E-02

6.81E+00

8.94E+00

5.57E+05

8.77E+00

SD

7.36E-02

3.18E-01

4.94E-02

7.71E+00

6.51E+00

1.92E+06

8.05E+00

f14

Mean

1.11E-04

6.42E+01

4.33E+01

5.56E-01

3.51E-02

2.29E+02

3.18E+01

SD

9.52E-05

7.66E+01

8.17E+01

1.23E-01

1.91E-01

7.53E+01

7.47E+01

f15

Mean

5.94E+01

2.25E+02

1.54E+02

2.58E+02

3.04E+02

2.98E+02

3.15E+02

SD

6.31E+01

1.27E+02

1.31E+02

1.45E+02

1.42E+02

8.53E+01

1.59E+02

f16

Mean

2.82E+01

5.27E+02

8.77E+01

2.18E+02

1.77E+02

1.68E+03

1.27E+02

SD

4.63E+01

2.92E+02

7.71E+01

1.19E+02

1.49E+02

5.78E+01

9.41E+01

f17

Mean

9.08E+02

9.52E+02

8.32E+02

9.48E+02

1.02E+03

5.37E+02

1.08E+03

SD

5.10E+00

2.70E+01

2.20E+01

3.88E+01

2.74E+01

7.75E+01

1.61E+02

f18

Mean

2.48E+02

3.04E+02

3.48E+02

4.28E+02

2.52E+02

5.35E+02

4.31E+02

SD

8.96E+01

1.41E+02

1.47E+02

7.85E+01

1.94E+02

8.02E+01

1.07E+02

f19

Mean

2.78E+02

4.69E+02

3.83E+02

8.27E+02

3.56E+02

7.04E+02

3.73E+02

SD

8.05E+01

1.04E+02

6.27E+01

1.03E+02

1.08E+02

1.49E+02

1.18E+02

All codes and source files are available at https://aliasgharheidari.com/INFO.html

18

5.1. Assessment of the exploitative behavior

In this section, the exploitation ability of INFO is investigated using the UFs.

Functions f1- f7 are unimodal and have one global solution. Table 5 reveals that INFO is very

promising and competitive with the comparative algorithms. Specifically, it was the best

method to optimize all functions in terms of the average objective function values for 30 runs.

The proposed algorithm can outperform the others on all functions according to standard

deviation values, except for GWO on function f5. Therefore, INFO can afford appropriate

exploitation search ability due to the embedded exploitation phase.

5.2. Assessment of the exploratory behavior

To inspect the exploration search capability of INFO in the study, MFs (f8-f13) were

used. These functions have many local optima solutions whose number rises exponentially

with the dimension of the problems and, thus, are suitable to verify optimizers' exploration

search ability. Table 5 presents the results of INFO and other optimization methods, revealing

that the proposed algorithm for MFs presents a very suitable exploration search ability.

Specifically, INFO outperformed all other algorithms in terms of the average of the objective

function and standard deviation values found for all the functions, except on the standard

deviations of functions f8 and f13. The presented results demonstrate that the INFO algorithm

has an excellent competency in exploration search.

5.3. Assessment of ability escaping from local optimum

In this section, the ability of INFO on CFs (f14-f19) is examined. These functions are

very challenging for optimizers because only an appropriate balance between exploitation and

exploration can escape local optimum solutions. Table 5 displays the results of all examined

algorithms on the CFs. It is evident that INFO achieved favorable results regarding the

average of objective function values for all functions and lower standard deviations for

functions f14-f17 compared to the other algorithms but failed on functions f18 and f19. This means

that the proposed algorithm has a suitable balance between exploitation and exploration,

preventing it from getting stuck in high local optimum solutions. This efficient performance

is due to the obtained vectors by using the updating rule and vector combining operators. The

updating rule provides two vectors to improve the local (exploitation) and global (exploration)

search in the search space (Eq. 8). Then, the vector combining operator combines them with

the current vector with a certain probability. This process helps the INFO algorithm explore

the search space on both the global and local scales with a suitable balance. Also, the local

operator makes the optimization process safe from trapping in local optima positions.

By employing the Friedman test [50], it was found that the INFO algorithm achieved

a top rank, followed by GWO and GSA, as seen in Table 6. This further verifies that INFO’s

performance is better than the other well-known optimizers.

All codes and source files are available at https://aliasgharheidari.com/INFO.html

19

Table 6. Mean rankings computed by Friedman test for test functions

Function

INFO

GWO

GSA

SCA

PSO

BA

GA

f1

1

2

3

7

5

4

6

f2

1

2

3

4

7

5

6

f3

1

2

5

7

4

3

6

f4

1

2

5

7

4

3

6

f5

1

2

4

7

5

3

6

f6

1

4

6

7

3

2

5

f7

1

2

4

6

7

3

5

f8

1

4

7

6

3

5

2

f9

1

2

5

6

7

3

4

f10

1

2

3

7

5

6

4

f11

1

2

7

6

3

4

5

f12

1

3

6

7

2

4

5

f13

1

3

6

7

2

5

4

f14

1

6

2

7

5

4

3

f15

1

3

6

5

2

7

4

f16

1

6

4

7

2

3

5

f17

3

5

6

1

2

7

4

f18

1

3

2

7

4

6

5

f19

1

5

4

7

2

6

3

Mean Rank

1.11

3.16

4.63

6.21

3.89

4.37

4.63

Final Rank

1

2

3

5

4

6

4

5.4. Investigation of convergence speed

In this paper, three metrics, including search history, trajectory curve, and convergence

rate, were considered to assess the INFO algorithm's convergence behavior. Accordingly, 8

different benchmark functions, i.e. f1, f3, f6, f7, f9, f10, f11, and f13, each with a dimension of 2, were

chosen. To solve these test functions, INFO used five solutions over 200 iterations.

The search history and trajectory curves of the five solutions in their first dimension

are depicted in Fig. 5. Generally, a low-density distribution illustrates the exploration, and a

high-density distribution represents the exploitation. According to this figure, the solutions'

distribution density demonstrates how INFO can search globally and locally in the solution

space, where the solutions have a high density in the region close to the global optima and

have a low density in the regions far from the global optima. Therefore, it can be concluded

that INFO can successfully explore promising regions in the solution space to explore the

best position.

All codes and source files are available at https://aliasgharheidari.com/INFO.html

21

Fig. 5. Search history, trajectory, average fitness, and convergence metrics

In the average fitness graph in Fig. 5, the varied history of the fitness of solutions in

INFO during the optimization process can be seen, where the average fitness curve suddenly

decreased in the early iterations. This behavior confirms the superior convergence speed and

accurate search capability of INFO.

The trajectory curves represent the global and local search behaviors of each

optimizer. Fig. 5 displays the trajectory graphs of five solutions for the first dimension,

revealing the high variation of curves in the initial generation. As the number of iterations

increased, the variation of the curves decreased. Since high and low variations indicate the

exploration and exploitation, respectively, it can be deduced that the INFO algorithm first

performs the exploration and then the exploitation.

All codes and source files are available at https://aliasgharheidari.com/INFO.html

22

The most crucial goal of each optimization algorithm is to obtain global optima, which

can be achieved by obtaining the convergence curves to visualize the algorithms' behaviors.

According to Fig. 5, the convergence variations of functions f1, f6, f9, f11, and f13 dropped rapidly

in the early iterations, demonstrating that INFO implemented the exploration search more

effectively than the exploration. In opposition, the convergence variations of functions f3, f7,

and f10 decreased relatively slow, indicating the better efficiency of INFO in the exploration

search than the exploitation.

Different variations of convergence graphs for the optimizers are displayed in Fig. 6,

which compares the convergence speeds of the INFO algorithms and other optimizers on

some of the benchmark functions. Accordingly, INFO can compete on the same level as the

other contestant algorithms with a very promising convergence speed.

As first reported by Berg et al. [51], sudden changes in the convergence curve during

the early optimization steps effectively reach the best solution. This behavior helps an

optimization method to better search the search space, whereby the optimization process

should be decreased to support exploitation search in the end stages of the optimization

process. According to this viewpoint, during the optimization procedure, the exploitation

phase assists in probing the search space, and then the solutions converge to the best solution

in the exploration phase.

As can be observed in Fig. 6, INFO presents two convergence modes for optimizing

the problems. The first mode is rapid convergence in the initial iterations, whereby the INFO

algorithm can reach a more accurate solution than the contestant algorithms. This high

convergence rate can be seen in f1, f2, f4, f10, and f13. The second convergence mode tends to

increase the convergence rate by increasing the number of iterations. This is owed to the

proposed adaptation method in INFO that aids it to explore appropriate areas of the search

domain in the primary iterations and improves the convergence speed after almost 100

iterations. Regarding functions f7, f15 to f19 as the most complex and challenging benchmark

functions, the optimization results of these problems indicate that INFO profits from a

suitable balance of exploitation and exploration that helps it achieve the global solution.

Consequently, these results demonstrate the efficient performance of INFO to optimize

complex problems.

All codes and source files are available at https://aliasgharheidari.com/INFO.html

24

5.5. Wall-clock time analysis of INFO

This section investigates the run-time of INFO compared with other methods on 13

benchmark functions. All optimization methods were run ten times on each test function

separately, and the results are reported in Table 7. From the table, the INFO optimization

process took a relatively long time due to the calculation of its two operators (i.e., vector

combining and local search stage). Nevertheless, INFO outperformed some methods, such

as GA and GSA. Generally, albeit with a relatively time-consuming run-time, INFO has

considerable advantages over the other methods.

Table 7 Comparison of the run-time of INFO and other methods

INFO

GA

GSA

GWO

PSO

SCA

BA

f1

4.70

5.84

8.46

1.30

0.64

1.31

1.56

f2

4.70

5.96

7.18

1.42

0.77

1.14

1.67

f3

7.91

9.34

10.47

4.72

4.51

2.50

5.42

f4

4.08

5.04

7.06

1.18

0.52

1.13

1.55

f5

4.24

5.51

7.23

1.38

0.66

1.41

1.73

f6

4.07

5.05

7.03

1.20

0.50

1.52

1.39

f7

4.77

5.90

7.66

1.87

1.18

2.40

2.09

f8

4.38

5.75

7.28

1.44

0.81

1.80

1.83

f9

4.33

5.15

7.08

1.38

0.62

1.61

1.82

f10

4.14

5.36

7.17

1.65

0.76

1.78

1.76

f11

4.56

5.75

7.55

1.44

0.88

2.20

2.22

f12

6.50

8.04

9.38

3.37

2.80

4.59

4.28

f13

6.44

8.24

9.07

3.34

2.79

4.86

4.05

5.6. Assessment of INFO on CEC-BC-2017 test functions

To further evaluate the INFO algorithm, widely-used and complicated CEC-BC-2017,

benchmark problems were utilized, including rotated and shifted unimodal, multimodal,

hybrid, and composite test functions [52]. The characteristics of these problems are available

in Appendix A (Table 8). The mathematical formulation of these test functions is also available

in the initial IEEE report. The proposed INFO algorithm was assessed against these

benchmark functions, and its results were compared to those of other well-known optimizers.

For all test functions, the dimension was equal to 10. The optimizers were running 30 times

with 1000 iterations for each test function. The control parameters for each optimizer were

the same as those considered in Section 5. Table 9 reports the results acquired by INFO and

the other algorithms, including each function's average and standard deviation over 30 runs.

According to Table 10, INFO takes the top Friedman mean rank, and the efficiency

of the INFO, PSO, and GWO are much better than the other optimizers. This performance

illustrates INFO’s capability to outperform other well-known optimizers and further indicates

that INFO can solve complex optimization problems.

All codes and source files are available at https://aliasgharheidari.com/INFO.html

25

Table 8. Properties of the CEC-BC-2017 test functions [52]

Type

No.

Functions

Global

Domain

Unimodal

Function

f1

Shifted and Rotated Bent Cigar Function

100

[-100,100]

f3

Shifted and Rotated Zakharov Function

300

[-100,100]

Multimodal

Functions

f4

Shifted and Rotated Rosenbrock’s Function

400

[-100,100]

f5

Shifted and Rotated Rastrigin’s Function

500

[-100,100]

f6

Shifted and Rotated Expanded Scaffer’s Function

600

[-100,100]

f7

Shifted and Rotated Lunacek Bi_Rastrigin Function

700

[-100,100]

f8

Shifted and Rotated Non-Continuous Rastrigin’s

Function

800

[-100,100]

f9

Shifted and Rotated Levy Function

900

[-100,100]

f10

Shifted and Rotated Schwefel’s Function

1000

[-100,100]

Hybrid

Functions

f11

Hybrid Function of Zakharov, Rosenbrock and

Rastrigin’s

1100

[-100,100]

f12

Hybrid Function of High Conditioned Elliptic,

Modified Schwefel and Bent Cigar

1200

[-100,100]

f13

Hybrid Function of Bent Cigar, Rosenbrock and

Lunache Bi-Rastrigin

1300

[-100,100]

f14

Hybrid Function of Eliptic, Ackley, Schaffer and

Rastrigin

1400

[-100,100]

f15

Hybrid Function of Bent Cigar, HGBat, Rastrigin and

Rosenbrock

1500

[-100,100]

f16

Hybrid Function of Expanded Schaffer, HGBat,

Rosenbrock and Modified Schwefel

1600

[-100,100]

f17

Hybrid Function of Katsuura, Ackley, Expanded

Griewank plus Rosenbrock, Modifed Schwefel and

Rastrigin

1700

[-100,100]

f18

Hybrid Function of high conditioned Elliptic, Ackley,

Rastrigin, HGBat and Discus

1800

[-100,100]

f19

Hybrid Function of Bent Cigar, Rastrigin, Expanded

Grienwank plus Rosenbrock,

Weierstrass and expanded Schaffer

1900

[-100,100]

f20

Hybrid Function of Happycat, Katsuura, Ackley,

Rastrigin, Modified Schwefel and

Schaffer

2000

[-100,100]

Composition

Functions

f21

Composition Function of Rosenbrock, High

Conditioned Elliptic and Rastrigin

2100

[-100,100]

f22

Composition Function of Rastrigin’s, Griewank’s and

Modified Schwefel's

2200

[-100,100]

f23

Composition Function of Rosenbrock, Ackley,

Modified Schwefel and Rastrigin

2300

[-100,100]

f24

Composition Function of Ackley, High Conditioned

Elliptic, Girewank and Rastrigin

2400

[-100,100]

f25

Composition Function of Rastrigin, Happycat, Ackley,

Discus and Rosenbrock

2500

[-100,100]

f26

Composition Function of Expanded Scaffer, Modified

Schwefel, Griewank,

Rosenbrock and Rastrigin

2600

[-100,100]

f27

Composition Function of HGBat, Rastrigin, Modified

Schwefel, Bent-Cigar, High

Conditioned Elliptic and Expanded Scaffer

2700

[-100,100]

f28

Composition Function of Ackley, Griewank, Discus,

Rosenbrock, HappyCat, Expanded Scaffer

2800

[-100,100]

f29

Composition Function of shifted and rotated Rastrigin,

Expanded Scaffer and Lunacek Bi-Rastrigin

2900

[-100,100]

f30

Composition Function of shifted and rotated Rastrigin,

Non-Continuous Rastrigin and Levy Function

3000

[-100,100]

All codes and source files are available at https://aliasgharheidari.com/INFO.html

26

Table 9. Statistical results and comparison for CEC-BC- 2017 functions

Function

INFO

GWO

GSA

SCA

PSO

BA

GA

f1

Mean

1.00E+02

3.24E+07

3.89E+02

7.32E+08

1.87E+03

1.24E+09

1.64E+03

SD

2.39E-05

1.10E+08

4.21E+02

2.91E+08

2.46E+03

1.10E+09

1.30E+03

f3

Mean

3.00E+02

1.69E+03

1.06E+04

1.97E+03

3.00E+02

1.29E+04

2.30E+03

SD

2.04E-09

1.91E+03

1.96E+03

1.26E+03

4.60E-14

1.27E+04

1.31E+03

f4

Mean

4.00E+02

4.14E+02

4.06E+02

4.47E+02

4.05E+02

5.25E+02

4.09E+02

SD

5.33E-01

1.59E+01

5.58E-01

1.81E+01

1.21E+01

9.19E+01

1.75E+01

f5

Mean

5.12E+02

5.16E+02

5.58E+02

5.52E+02

5.36E+02

5.53E+02

5.40E+02

SD

6.08E+00

7.58E+00

1.10E+01

5.36E+00

1.34E+01

2.07E+01

1.27E+01

f6

Mean

6.00E+02

6.01E+02

6.23E+02

6.17E+02

6.07E+02

6.41E+02

6.20E+02

SD

6.65E-03

1.23E+00

8.57E+00

2.91E+00

5.05E+00

1.34E+01

1.06E+01

f7

Mean

7.24E+02

7.29E+02

7.15E+02

7.73E+02

7.24E+02

7.84E+02

7.49E+02

SD

6.79E+00

8.89E+00

2.58E+00

8.89E+00

6.36E+00

3.14E+01

1.86E+01

f8

Mean

8.12E+02

8.15E+02

8.20E+02

8.38E+02

8.21E+02

8.36E+02

8.21E+02

SD

5.13E+00

6.98E+00

4.14E+00

7.22E+00

9.81E+00

1.32E+01

1.03E+01

f9

Mean

9.00E+02

9.08E+02

9.00E+02

9.97E+02

9.00E+02

1.65E+03

1.05E+03

SD

7.77E-01

1.69E+01

0.00E+00

3.04E+01

4.72E-14

4.53E+02

1.02E+02

f10

Mean

1.64E+03

1.65E+03

2.80E+03

2.24E+03

1.92E+03

2.28E+03

2.03E+03

SD

2.40E+02

2.43E+02

3.38E+02

2.27E+02

2.21E+02

3.13E+02

3.19E+02

f11

Mean

1.11E+03

1.12E+03

1.14E+03

1.20E+03

1.14E+03

1.83E+03

1.15E+03

SD

8.72E+00

1.43E+01

1.16E+01

4.33E+01

1.73E+01

6.76E+02

3.87E+01

f12

Mean

2.78E+03

7.09E+05

7.61E+05

1.70E+07

1.57E+04

1.36E+06

9.90E+05

SD

1.80E+03

8.27E+05

4.90E+05

1.24E+07

1.08E+04

1.99E+06

1.06E+06

f13

Mean

1.44E+03

1.14E+04

1.14E+04

2.51E+04

9.80E+03

1.79E+04

9.37E+03

SD

1.23E+02

8.43E+03

2.36E+03

2.15E+04

7.18E+03

1.63E+04

5.33E+03

f14

Mean

1.43E+03

3.03E+03

6.78E+03

1.70E+03

1.73E+03

2.46E+03

3.81E+03

SD

1.02E+01

1.79E+03

2.01E+03

6.61E+02

4.41E+02

1.22E+03

2.19E+03

f15

Mean

1.52E+03

3.68E+03

2.00E+04

2.25E+03

2.35E+03

4.03E+04

3.68E+03

SD

1.72E+01

2.54E+03

4.62E+03

5.90E+02

1.42E+03

5.84E+04

2.09E+03

f16

Mean

1.65E+03

1.76E+03

2.16E+03

1.73E+03

1.85E+03

2.02E+03

1.93E+03

SD

5.73E+01

1.31E+02

1.19E+02

4.71E+01

7.58E+01

1.70E+02

1.10E+02

f17

Mean

1.72E+03

1.76E+03

1.84E+03

1.78E+03

1.77E+03

1.92E+03

1.76E+03

SD

1.50E+01

3.59E+01

1.22E+02

2.00E+01

3.03E+01

1.36E+02

2.95E+01

f18

Mean

1.86E+03

2.51E+04

8.37E+03

1.07E+05

9.60E+03

1.83E+04

1.21E+04

SD

4.13E+01

1.31E+04

4.13E+03

7.17E+04

8.67E+03

1.66E+04

1.11E+04

f19

Mean

1.91E+03

1.50E+04

4.28E+04

6.75E+03

3.00E+03

2.35E+04

7.47E+03

SD

7.06E+00

4.70E+04

1.92E+04

5.98E+03

1.77E+03

2.97E+04

4.58E+03

f20

Mean

2.01E+03

2.06E+03

2.29E+03

2.10E+03

2.08E+03

2.23E+03

2.14E+03

SD

1.22E+01

4.99E+01

9.60E+01

3.12E+01

4.47E+01

1.16E+02

7.08E+01

All codes and source files are available at https://aliasgharheidari.com/INFO.html

27

Table 9 (continued). Statistical results and comparison for CEC-BC- 2017 functions

Function

INFO

GWO

GSA

SCA

PSO

BA

GA

f21

Mean

2.28E+03

2.30E+03

2.36E+03

2.25E+03

2.30E+03

2.32E+03

2.32E+03

SD

5.39E+01

3.57E+01

2.22E+01

6.29E+01

5.82E+01

5.82E+01

4.36E+01

f22

Mean

2.30E+03

2.33E+03

2.30E+03

2.36E+03

2.30E+03

2.63E+03

2.31E+03

SD

1.65E+01

1.12E+02

6.85E-11

3.83E+01

9.27E-01

1.45E+02

1.03E+01

f23

Mean

2.62E+03

2.62E+03

2.75E+03

2.66E+03

2.69E+03

2.66E+03

2.72E+03

SD

7.28E+00

9.60E+00

5.60E+01

6.79E+00

3.96E+01

2.37E+01

4.77E+01

f24

Mean

2.75E+03

2.75E+03

2.55E+03

2.78E+03

2.73E+03

2.77E+03

2.81E+03

SD

7.87E+00

1.13E+01

1.17E+02

3.92E+01

1.32E+02

8.47E+01

1.38E+02

f25

Mean

2.92E+03

2.94E+03

2.94E+03

2.96E+03

2.92E+03

3.01E+03

2.93E+03

SD

3.07E+01

2.40E+01

1.36E+01

2.14E+01

2.29E+01

5.10E+01

2.31E+01

f26

Mean

3.11E+03

3.06E+03

3.70E+03

3.07E+03

3.07E+03

3.42E+03

3.55E+03

SD

3.71E+02

3.16E+02

6.96E+02

3.48E+01

2.85E+02

3.99E+02

4.82E+02

f27

Mean

3.09E+03

3.09E+03

3.26E+03

3.10E+03

3.16E+03

3.13E+03

3.23E+03

SD

1.61E+00

2.39E+00

3.95E+01

1.43E+00

4.39E+01

3.48E+01

4.36E+01

f28

Mean

3.30E+03

3.36E+03

3.46E+03

3.29E+03

3.17E+03

3.43E+03

3.29E+03

SD

1.66E+02

9.23E+01

3.08E+01

6.58E+01

4.16E+01

1.14E+02

1.58E+02

f29

Mean

3.17E+03

3.19E+03

3.45E+03

3.23E+03

3.23E+03

3.38E+03

3.28E+03

SD

3.11E+01

3.14E+01

1.42E+02

3.23E+01

4.08E+01

1.12E+02

6.89E+01

f30

Mean

8.55E+04

9.19E+05

9.83E+05

1.04E+06

9.80E+03

2.18E+06

4.45E+05

SD

2.49E+05

1.08E+06

2.61E+05

8.05E+05

5.69E+03

3.14E+06

1.01E+06

5.7. Ranking analysis of INFO

In this section, multiple statistical tests, including Friedman's [50], Bonferroni-Dunn's

[53], and Holm’s tests [54], are considered to evaluate the difference between the efficiency of

INFO and the other optimizers in the test functions. To implement a trustworthy comparison,

this research divided the test functions into three groups: G1 (group 1) includes unimodal,

multimodal, and composite functions (Tables 2-4); G2 (group 2) consists of CEC-BC-2017

test functions (Table 8), and G3 (group 3) is the combination of G1 and G2.

All codes and source files are available at https://aliasgharheidari.com/INFO.html

28

Table 10. Mean rankings computed by Friedman test for CEC-BC-2017 functions

Function

INFO

GWO

GSA

SCA

PSO

BA

GA

f1

1

5

2

6

4

7

3

f3

1.5

3

6

4

1.5

7

5

f4

1

5

3

6

2

7

4

f5

1

2

7

5

3

6

4

f6

1

2

6

4

3

7

5

f7

2.5

4

1

6

2.5

7

5

f8

1

2

3

7

4.5

6

4.5

f9

2

4

2

5

2

7

6

f10

1

2

7

5

3

6

4

f11

1

2

3.5

6

3.5

7

5

f12

1

3

4

7

2

6

5

f13

1

4.5

4.5

7

3

6

2

f14

1

5

7

2

3

4

6

f15

1

4.5

6

2

3

7

4.5

f16

1

3

7

2

4

6

5

f17

1

2.5

6

5

4

7

2.5

f18

1

6

2

7

3

5

4

f19

1

5

7

3

2

6

4

f20

1

2

7

4

3

6

5

f21

2

3.5

7

1

3.5

5.5

5.5

f22

2

5

2

6

2

7

4

f23

1.5

1.5

7

3.5

5

3.5

6

f24

3.5

3.5

1

6

2

5

7

f25

1.5

4.5

4.5

6

1.5

7

3

f26

4

1

7

2.5

2.5

5

6

f27

1.5

1.5

7

3

5

4

6

f28

4

5

7

2.5

1

6

2.5

f29

1

2

7

3.5

3.5

6

5

f30

2

4

5

6

1

7

3

Mean Rank

1.55

3.38

5.02

4.59

2.86

6.07

4.53

Final Rank

1

3

6

5

2

7

4

In the multiple statistical tests, the optimizers' results were first investigated to

determine their equality. When inequality was found, post-hoc analysis was performed to find

out which optimizer’s performance is significantly different from INFO. Therefore, the

Friedman test was conducted once again to obtain the optimizers' average ranks on the three

groups, as shown in Fig. 7.

All codes and source files are available at https://aliasgharheidari.com/INFO.html

29

Fig. 7. Bonferroni-Dunn test for all optimizers and different groups

The Bonferroni-Dunn test is a post-hoc analysis used to determine if the efficiency of

two optimizers is significantly different and if the difference between mean ranks of

optimization methods is larger than the critical difference (CD):

All codes and source files are available at https://aliasgharheidari.com/INFO.html

30

( )

1

6

mm

CD Q n

+

=

(13)

where is the critical value, calculated based on work in [55], and and are the numbers

of optimizers and test problems, respectively. In this work, INFO was introduced as the

control optimizer. In Fig. 8, the horizontal lines show CD as the threshold for the INFO

algorithm. For two common significant levels of 0.05 and 0.1, the threshold lines were

determined and are displayed in Fig. 8 as dashed and dotted lines, respectively. In the three

groups, INFO had the lowest mean ranks (G1 = 1.11, G2 = 1.55, G3 = 1.33) and, thus, can

outperform the other optimizers, which have mean ranks above the CD lines. It is pertinent

to note that the PSO rank is below the threshold line in G2.

However, the Bonferroni-Dunn test does not determine the main difference between

the optimizers if their mean ranks are less than the threshold line. Therefore, the present

research used Holm’s test to specify whether there is a substantial difference between the

optimizers, with ranks less than the threshold line. To implement Holm’s test, all optimizers

were sorted based on their p-value and were compared with , where is the algorithm

number. If the p-value is less than the corresponding significant level (), the optimizer is

significantly different. Tables 11 and 13 (G1 and G3) show the Bonferroni-Dunn test results

for levels 0.05 and 0.1, revealing a significant difference between the efficiency of INFO

and the other optimizers. For G2 (Table 12), the Bonferroni-Dunn test indicates no significant

difference between INFO and PSO, while Holm’s test demonstrates a significant difference

between these two. Consequently, in Fig. 8, the mean ranks of INFO in the three groups are

very close to each other, while the other optimizers have unstable performance in various

groups. Finally, it may be concluded that INFO has a reliable and accurate efficiency in all

groups compared to the other optimizers.

Table 11. Holm’s test for G1 test functions (INFO is as the control optimizer)

INFO VS.

Rank

Z-value

P-value

(.)

(.)

SCA

6.21

7.276

1.71E-13

0.00833

0.01667

GA

4.63

5.022

2.55E-07

0.01000

0.02000

GSA

4.63

5.022

2.55E-07

0.01250

0.02500

BA

4.37

4.651

1.65E-06

0.01667

0.03333

PSO

3.89

3.966

3.64E-05

0.02500

0.05000

GWO

3.16

2.924

1.72E-03

0.05000

0.10000

Table 12. Holm’s test for G2 test functions (INFO is as the control optimizer)

INFO VS.

Rank

Z-value

P-value

(.)

(.)

BA

6.069

7.967

7.77E-16

0.00833

0.01667

GSA

5.017

6.116

4.78E-10

0.01000

0.02000

SCA

4.586

5.358

4.19E-08

0.01250

0.02500

GA

4.534

5.252

7.48E-08

0.01667

0.03333

GWO

3.379

3.225

6.28E-04

0.02500

0.05000

PSO

2.862

2.309

1.04E-02

0.05000

0.10000

All codes and source files are available at https://aliasgharheidari.com/INFO.html

31

Table 13. Holm’s test for G3 test functions (INFO is as the control optimizer)

INFO VS.

Rank

Z-value

P-value

(.)

(.)

BA

5.40

5.807

3.18E-09

0.00833

0.01667

SCA

5.22

5.550

1.42E-08

0.01000

0.02000

GSA

4.82

4.979

3.18E-07

0.01250

0.02500

GA

4.58

4.637

1.76E-06

0.01667

0.03333

PSO

3.38

2.924

1.72E-03

0.02500

0.05000

GWO

3.27

2.767

2.82E-03

0.05000

0.10000

5.8. Performance comparison of INFO with advanced algorithms

In this section, the performance of INFO is compared with advanced algorithms,

including SCADE [56], CGSCA [57], OBLGWO [58], RDWOA [59], CCMWOA [60],

BMWOA [61], CLPSO [62], RCBA [63], and CBA [64] on the CEC-BC-2017 test functions.

For all the optimization algorithms, the population size and the total number of generations

were set to 30 and 500, respectively. To decrease the effect of random behavior in each

optimizer on the results, all optimizers were run 30 different times for each test function.

Table 14 indicates the average (AVG) and standard deviation results obtained by all

optimization methods, which confirm that INFO is a very competitive optimizer to solve

CEC-BC-2017 functions. In f1, f3-f19, f21, f23, f24, f26, f29, and f30, the AVG of INFO was smaller

than that of the other optimizers. These results show that INFO performed better on 24 out

of 29 test functions than other advanced algorithms.

Furthermore, the Wilcoxon Signed Ranks (WSR) test [65] was utilized to compare all

optimizers' overall efficiency on 30 independent runs. In the WSR, + indicates the sum of

ranks during all runs in which INFO outperformed the competitor algorithm. Comparatively,

- represents the sum of ranks during all runs in which the competitor algorithm

outperformed INFO. -value specifies the significance of the results in a statistical hypothesis

test ( = 0.05). The comparisons of the optimizers by the WSR test are reported in Table 15,

where the symbol ‘+’ shows that INFO has better efficiency than its competitor algorithm; ‘-

’ indicates that the competitor algorithm’s efficiency is better than INFO; and ‘=’ denotes

similar performance between INFO and the competitor algorithm. Each test's statistical

results for the 30 runs are presented in Table 16, which shows that the INFO algorithm can

perform impressively better than its competitors.

Fig. 8 also depicts the convergence curve of some CEC-BC-2017 test functions.

According to this figure, INFO can explore a superior solution at a fast convergence rate

compared with the other optimizers. Moreover, the Friedman test was utilized to calculate all

algorithms' average ranks, revealing that INFO has the best rank value (1.22) and performed

much better than the other optimizers (Table 17).

All codes and source files are available at https://aliasgharheidari.com/INFO.html

32

Table 14. Statistical results and comparison for CEC-BC- 2017 functions

Function

INFO

SCADE

CGSCA

OBLGWO

RDWOA

CCMWOA

BMWOA

CLPSO

RCBA

CBA

f1

Mean

1.32E+05

2.85E+10

2.53E+10

1.69E+08

1.01E+09

1.17E+10

1.23E+09

1.23E+10

7.87E+05

1.99E+06

SD

2.15E+05

3.80E+09

3.85E+09

8.23E+07

1.35E+09

3.83E+09

5.66E+08

2.54E+09

2.65E+05

2.17E+06

f3

Mean

2.09E+04

7.72E+04

7.06E+04

5.09E+04

6.27E+04

7.36E+04

8.10E+04

1.57E+05

9.56E+04

1.01E+05

SD

8.46E+03

5.86E+03

8.92E+03

8.86E+03

1.21E+04

5.19E+03

7.01E+03

2.14E+04

4.39E+04

6.20E+04

f4

Mean

5.00E+02

6.12E+03

3.46E+03

5.49E+02

6.39E+02

1.78E+03

7.29E+02

3.02E+03

5.08E+02

5.14E+02

SD

2.78E+01

1.24E+03

1.15E+03

2.56E+01

1.18E+02

7.37E+02

7.13E+01

9.93E+02

2.58E+01

2.78E+01

f5

Mean

6.15E+02

8.79E+02

8.53E+02

7.00E+02

7.32E+02

8.05E+02

8.26E+02

8.26E+02

8.20E+02

8.12E+02

SD

2.92E+01

2.11E+01

2.63E+01

5.54E+01

6.57E+01

3.78E+01

3.01E+01

2.12E+01

5.21E+01

6.58E+01

f6

Mean

6.13E+02

6.80E+02

6.71E+02

6.33E+02

6.36E+02

6.71E+02

6.68E+02

6.58E+02

6.73E+02

6.77E+02