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Giza Pyramids Construction an ancient‑inspired metaheuristic .zip （2个子文件）

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Evolutionary Intelligence

https://doi.org/10.1007/s12065-020-00451-3

RESEARCH PAPER

Giza Pyramids Construction: anancient‑inspired metaheuristic

algorithm foroptimization

SasanHari

· JavadMohammadzadeh

· MadjidKhalilian

· SadoullahEbrahimnejad

Received: 4 March 2020 / Revised: 25 April 2020 / Accepted: 2 July 2020

Abstract

Nowadays, many optimization issues around us cannot be solved by precise methods or that cannot be solved in a reasonable

time. One way to solve such problems is to use metaheuristic algorithms. Metaheuristic algorithms try to ﬁnd the best solution

out of all possible solutions in the shortest time possible. Speed in convergence, accuracy, and problem-solving ability at high

dimensions are characteristics of a good metaheuristic algorithm. This paper presents a new population-based metaheuristic

algorithm inspired by a new source of inspiration. This algorithm is called Giza Pyramids Construction(GPC) inspired by

the ancient past has the characteristics of a good metaheuristic algorithm to deal with many issues. The ancient-inspired is

to observe and reﬂect on the legacy of the ancient past to understand the optimal methods, technologies, and strategies of

that era. The proposed algorithm is controlled by the movements of the workers and pushing the stone blocks on the ramp.

This algorithm is compared with ﬁve standard and popular metaheuristic algorithms. For this purpose, thirty diﬀerent and

diverse benchmark test functions are utilized. The proposed algorithm is also tested on high-dimensional benchmark test

functions and is used as an application in image segmentation. The results show that the proposed algorithm is better than

other metaheuristic algorithms and it is successful in solving high-dimensional problems, especially image segmentation.

Keywords Metaheuristic· Optimization· Giza Pyramids Construction algorithm· GPC algorithm· Ancient-inspired·

High-dimensional tests· Image segmentation· Benchmark test functions

1 Introduction

Optimization applications are numerous. Each process has

the potential to be optimized. There are no companies and

institutions that not involved in optimization. Many challeng-

ing applications in science and technology can be formulated

as an optimization problem. Optimization can reduce time,

cost and risk or increase proﬁt, quality, and eﬃciency. In

the industry, for example, there are cost and service quality

optimization. There are also many ways to optimize time

for product planning. There are many optimization issues in

science, engineering, economics and business that are dif-

ﬁcult to solve [1]. They are not solved accurately and within

a reasonable time. Using the approximation algorithm is the

main option to solve these problems.

Approximate algorithms are classiﬁed into two catego-

ries: heuristic and metaheuristic. Heuristics depend on the

type of problem. They are usually designed and used for

speciﬁc issues. Metaheuristics are more popular and are used

for many issues [2]. They can be used for almost any kind of

problem. The metaheuristic solves problems that seem to be

diﬃcult, by searching for a large space of solutions. These

algorithms achieve this goal by eﬀectively exploring space

and reducing the size of the solution space. Metaheuris-

tics solve problems faster, solve bigger problems, and are

stronger algorithms. They are also very ﬂexible and easy to

design and implement.

The metaheuristic is a branch of optimization in computer

science and applied mathematics that deals with complex

* Sasan Hariﬁ

s.hariﬁ@kiau.ac.ir

Javad Mohammadzadeh

j.mohammadzadeh@kiau.ac.ir

Madjid Khalilian

khalilian@kiau.ac.ir

Sadoullah Ebrahimnejad

ibrahimnejad@kiau.ac.ir

Department ofComputer Engineering, Karaj Branch,

Islamic Azad University, Karaj, Iran

Department ofIndustrial Engineering, Karaj Branch, Islamic

Azad University, Karaj, Iran

Evolutionary Intelligence

1 3

computation theory and algorithms. It also covers other

areas such as artiﬁcial intelligence, computational intel-

ligence, soft computing, mathematical programming, and

operations research. In practice, metaheuristics have been

very eﬀective in solving complex real-world problems, and

have also played a signiﬁcant role in reducing costs and

expanding rapidly in various ﬁelds [3].

The metaheuristics include the following categories.

These are Evolutionary-based, Trajectory-based, and Nature-

inspired methods. We do not claim that this categorization

is unique. Some algorithms can fall into several diﬀerent

categories.

Evolutionary models are an abstract model of biological

evolution in which populations that are considered candi-

dates for a solution are frequently exposed to natural selec-

tion or genetic diversity [4]. Evolutionary is modeled on the

concept of competition. This type of metaheuristics simu-

lates species evolution. They are based on the evolution of

the population. Populations are usually randomly generated,

and each individual is actually a solution. An objective func-

tion determines which solution is appropriate. At each stage

according to the pattern of choice, anyone who has better

suitability is selected with a higher probability. Then the

population selected are reproduced by diﬀerent operators

to create a new generation of oﬀspring. These operators are

usually crossover and mutation. Finally, an alternative plan

is applied to determine which of the oﬀspring and parents

will remain. Some of the popular evolutionary-based tech-

niques are Genetic Algorithm (GA) [5], Memetic Algorithm

(MA) [6], Diﬀerential Evolution (DE) [7], Harmony Search

(HS) [8], Clonal Selection Algorithm (CSA) [9], and so on.

Trajectory-based are methods that work on a solution.

In this way, the problem space is searched with a path [10].

The characteristics of these methods are usually determined

by the type of problem. Trajectory-based metaheuristics

improve a single solution. These are carried out with repeti-

tive routines through which they move from one solution to

another. It has shown its eﬀectiveness in a variety of issues.

Some of the popular techniques in this category are Simu-

lated Annealing (SA) [11], Tabu Search (TS) [10], Iterated

Local Search (ILS) [12], Guided Local Search (GLS) [13],

Greedy Randomized Adaptive Search Procedure (GRASP)

[14], Variable Neighborhood Search (VNS) [15], and so on.

Nature-inspired are methods that follow the laws of

nature. Nature has simple and understandable laws. In fact,

the collective behavior of most organisms is as a seeker in

the problem space that leads to the goal and solution [16].

Nature-inspired itself contains Swarm-based, Bio-inspired,

Physics/Chemistry-based, Human-based, and Plant-based

methods. Swarm-based is about group behaviors and swarm

intelligence of a set of beings. Swarm can be deﬁned as an

organized set of agents or entities that work together. It can

be an example of the lives of ants, bees, termites, ﬁsh, and

birds. Many swarm-based algorithms are presented such as

Particle Swarm Optimization (PSO) [17], Fireﬂy Algorithm

(FA) [18], Artiﬁcial Bee Colony (ABC) [19], Ant Colony

Optimization (ACO) [20], Emperor Penguins Colony (EPC)

[21], and so on. It is true that swarm intelligence algorithms

can be considered as a subset of bio-inspired algorithms.

But many bio-inspired algorithms do not directly use swarm

behavior. So it is better to put them in a separate subset

namely Bio-inspired. The Krill Herd Algorithm (KHA)

[22], Crow Search Algorithm (CSA) [23], Grey Wolf Opti-

mizer (GWO) [24], Owl Search Algorithm (OSA) [25], Ant

Lion Optimizer (ALO) [26], and so on are examples of bio-

inspired algorithms. As mentioned earlier, sometimes algo-

rithms can be in several categories. Physics/Chemistry-based

are mostly created by imitating the laws of physics or chem-

istry. These features and laws include electrical charge, grav-

ity, physical and chemical changes of materials, and so on.

From this subset we can refer to Chemical Reaction Opti-

mization (CRO) [27], Black Hole (BH) [28], Multi-Verse

Optimizer (MVO) [29], Thermal Exchange Optimization

(TEO) [30], and so on. Human-based methods models all

the individual and social behaviors of human. This subset

includes individual behaviors such as how humans search

for and adapt to the environment, and how human beings

behave with diﬀerent emotions. Social behaviors, such as

the chaos that occurs in society, the way human beings work

together or even imperialist can also be modeled. The most

popular algorithms in this subset are Imperialist Competi-

tive Algorithm (ICA) [31], Cultural Algorithm (CA) [32],

Teaching–Learning-Based Optimization (TLBO) [33], and

recently published Political Optimizer (PO) [34]. Plant-

based methods are inspired by the process of plant growth,

plant dispersal, root and plant expanding, and so on. In gen-

eral, any algorithm that somehow models a plant falls into

this subset. Example of algorithm in this subset are Invasive

Weed Optimization (IWO) [35], and Artiﬁcial Root Forag-

ing Algorithm (ARFA) [36].

The ancient past is the new ideology and origin of inspi-

ration that introduced by this paper for the ﬁrst time. In

the ancient past there were numerous limitations, but vari-

ous man-made structures show that the limitations and the

lack of hardware and software facilities have given rise

to some sort of optimization in ancient times. The Giza

Pyramids Construction

(GPC) optimization algorithm is

also presented as the ﬁrst ancient-inspired metaheuristic

algorithm. Figure1 shows the classiﬁcation of metaheuris-

tics explained in this paper. In this ﬁgure, the existing clas-

siﬁcations and the position of the new ancient-inspired

Source codes of GPC are publicly available at www.hariﬁ .com.

Evolutionary Intelligence

1 3

category are shown. This new inspiration can include all

the features of the existing inspirations.

The main contribution and innovation of this paper

is the use of a novel and eﬃcient source of inspiration

compared to existing sources of inspiration. This novel

source of inspiration is actually a combination of evolu-

tion and nature with many new features that the existence

of limitation in the ancient past has led to the ﬂourishing

of its capabilities. The main goal of this study is to pro-

vide a new metaheuristic algorithm based on this novel

source of inspiration, as well as to test and apply it to a

speciﬁc application to show that this approach is feasible

and reliable.

As mentioned earlier, metaheuristics are used in almost

every ﬁeld. These ﬁelds range from computer science [2] to

industry [37] or civil engineering [38]. Their applications

are also varied, for example in computer science they can

be used to optimize fuzzy systems [39], feature selection

[40, 41], information retrieval [42], and so on. In this paper

as an application, we decided to apply the proposed algo-

rithm in image segmentation. Image segmentation is the

process of dividing an image into homogeneous areas can

be considered a high-dimensional problem. Segmentation

is important because it prepares the image for applications

such as machine vision, and image analysis including medi-

cal images, satellite images, and so on.

Therefore, the experiments performed in this paper are

divided into two parts. The ﬁrst is related to the standard

experiment using benchmark test functions, which are per-

formed in two ways: low-dimensions and high-dimensions.

The second is related to the experiment of application in

image segmentation, which is itself an example of a high-

dimensional problem.

The remainder of this paper is structured as follows:

Sect.2 introduces ancient-inspired ideology. Section3

describes Giza Pyramids Construction (GPC) algorithm.

Section4 includes experimental results and discussion. Sec-

tion5 provides the statistical analysis. Section6 represents

high-dimensional tests. Section7 presents application in

image segmentation. Finally, Sect.8 represents conclusions.

2 Ancient‑inspired ideology

Ancient refers to past events from the beginning of the writ-

ing and recording of the history of humanity to the begin-

ning of the post-classical period [43, 44]. The length of this

period is approximately 5000years. Historians consider

the end of this period to be 500 AD. There are two ways

to better understand this era. The ﬁrst is through archeol-

ogy. Archeology is the exploration and study of ancient

artifacts to interpret and reconstruct human past behavior

[45]. Archaeologists are searching the ruins of ancient cit-

ies to ﬁnd clues about the lifestyle and the time period [46].

The second way is through studying textual sources. Textual

sources are narratives of ancient historians [47]. Many of the

events described by them are based on an understanding of

ancient history.

In the ancient past, productive work enabled humans to gain

wealth. In fact, work was a religious duty and a moral virtue in

public morality. The technical skills were highly commended.

Work and technical skills were the core of human intelligence

Fig. 1 Classiﬁcation of metaheuristic algorithms

Evolutionary Intelligence

1 3

and the key to understanding nature. This core and this key

gradually led to the emergence of superior technologies in the

ancient past [48]. One of the most attractive parts of ancient

times is the study of ancient technology and science. In history,

technology and ancient science evolved during the develop-

ment of ancient civilizations and technological advances in

engineering [49]. Among the technologies can be mentioned

such as Egyptians, Indians, Chinese, Greeks, Romans and Ira-

nians technologies.

The fact is that the advanced civilizations mentioned cre-

ated the knowledge infrastructures on a large scale so that

these infrastructures were not found elsewhere. This knowl-

edge infrastructure has evolved into an integrated network with

the spread of civilizations, where science and technology have

been easily circulated. In these civilizations, the elites have

always strived to acquire each other’s technical knowledge

[50]. An essential element in the development of antiquity

construction has been the use of pre-manufactured standard-

size materials. This greatly reduced construction costs and

improved development time. In ancient Egyptian civilization,

for example, stones were brought to the site to prepare the

pyramids [50]. Technology in the ancient past was always

used to optimize and reduce costs. Certainly, inventions such

as concrete and standard building materials would deﬁnitely

cost less for construction projects and give more freedom to

the architects of the structural design. Also, innovations in

construction could create large-scale complex structures with

low cost. Standardization of building materials and the crea-

tion of manufacturing technologies could even reduce labor

costs [50].

In general, the emergence of construction techniques and

the use of new strategies by builders has been a major innova-

tion that has had a profound impact on the social, economic

and cultural history of ancient civilizations. The question,

however, is how the workforce management strategy has been

and how it has been used. Looking at some of the monuments

and artifacts from antiquity and estimating the time of con-

struction of these monuments and considering the emergence

of new technology and the lack of advanced technology at that

time, it raises the question that how was the optimization strat-

egies in construction these buildings and monuments [50]. The

ancient-inspired ideology is the observation and thinking about

the remnants and legacies of antiquity and the understanding

of the management and strategy used in that era to advance

civilization and optimize and reduce the cost of living.

3 Giza Pyramids Construction (GPC)

algorithm

The Giza Pyramids complex, also known as Giza Necropo-

lis, is a site that has three large pyramids, all built during the

fourth dynasty of ancient Egypt [51]. The largest pyramid,

also known as the Seven Wonders, is called the Khufu Pyra-

mid. The other two pyramids are named Khafre and Men-

kaure [52]. According to archaeologists, the method of con-

struction is diﬀerent from each other because construction

of pyramids have been developed over time. How to manage

the workers was the most important issue in building the

pyramids. Due to the lack of hardware facilities, relatively

short construction time and the large number of stone blocks

used in the pyramids, its construction has been optimized.

In this section, the construction method, the inspiration, and

the proposed algorithm are described in detail.

3.1 The construction

Many theories have been put forward about the pyramids

construction methods, none of which is hundred percent

approved. Many believe that the stones of this pyramids

were removed from mines, shipped and then placed in

place. Such a way that, they used ramps to mount them at

higher levels [53, 54]. The Greeks believed that slaves were

exploited in the construction of the pyramids, but new ﬁnd-

ings suggest that the creators of the pyramids were skilled

workers who lived well in the surrounding villages. The

labor force needed to build the pyramids was an average of

fourteen thousand people and a maximum of forty thousand

[55]. Without the use of metal tools, wheels and other tools,

these pyramids were built over a period of 10–20years. The

number of blocks used in the largest pyramid is two mil-

lion pieces. The workers, including slaves, coolies, masons,

metalworkers, carpenters, and foreman.

In addition to the challenges of project construction, there

is a need for an advanced approach to project management,

construction management and method of construction. The

project involved building a prominent construction plant.

Issues such as feeding, housing, the payment of workers’

salaries, and work scheduling for timely completion before

the death of the pharaoh were also considered [56]. The

pyramid stands today as awesome testimony to the skill and

sheer determination of the ancient race that built it. The

complexity and logistical requirements of this project are

simply extraordinary.

3.2 The inspiration

As noted, the workers are slaves, coolies, masons, metal-

workers, carpenters, led by an expert agent. This expert

agent is a foreman called Pharaoh’s special agent. Workers

carry stone blocks. This is done under the direct supervi-

sion of the Pharaoh’s agent. There may be diﬀerent workers

each responsible for carrying a stone block. The task report

should be regularly given to Pharaoh’s special agent. This

Evolutionary Intelligence

1 3

task is performed by each worker. Figure2 illustrates an idea

of how these factors do the job, namely the simple worker,

and the Pharaoh’s special agent.

The workers at the construction site each have a position.

A worker who does his job better will receive sublime rank

as a reward. So there is a competition to get sublime rank.

The best rank is related to Pharaoh’s special agent. Also,

the energy lost during the stone block transport phase may

cause workers to rest for some time. If a worker loses too

much power or gets tired, he will be substituted by young

blood and energetic workers. This means that if there is no

improvement in the workforce, it must be substituted with a

new one. The weak worker may be more eﬃcient elsewhere.

In addition to competing for sublime rank, there is another

motivational competition among workers as each can gain

experience and expertise.

During the pyramid construction process, stone blocks

that are scattered by agents such as miners around the con-

struction site are carried by the workers to the construction

site. This is done every day so that the stone blocks are col-

lected from around and dragged to the pyramid. Ramps were

used to build the pyramid. The distance between the stone

block and the location of its installation in the pyramid must

be traveled. The distance traveled to push the stone is meas-

ured by the ability of the workers. During the workday, if

enough power is available, more shipment will be carried out

by the workers, making the block closer to the installation

site in the pyramid. It should be noted, the ramp gradient,

initial velocity, and friction force inﬂuence the movement

of the stone block.

3.3 The proposed algorithm

Suppose that stone blocks are scattered around the con-

struction site. Workers have to push the stone blocks to

the installation place. The initial position of each block

and its cost are known. Ramps are used to move the stone

blocks to the installation place. The slope of the ramp and

its friction affect the displacement of the stone blocks.

Also, the workers themselves are constantly moving to

find the best position to dominate the stone block. Due to

the different characteristics of each worker, it is possible

to substitute workers to balance the power of the workers

for carrying a stone block. Therefore, the position of some

workers will be substituted with others. This substituting

Fig. 2 Subjective perception of Pharaoh’s special agent and workers

Fig. 3 Flowchart of the proposed GPC

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