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Abstract

In order to effectively strengthen the exploration and exploitation capabilities of the arithmetic optimization algorithm (AOA) and the balance of search ability between the two is realized, a novel mathematical operator-based arithmetic optimization algorithm (MAOA) is proposed. Firstly, the exploitation and exploration abilities of the population are improved through mathematical symmetry operators and median operators, respectively. Secondly, the balance between exploration and exploitation of AOA algorithm is effectively strengthened by using sine–cosine operator. Finally, the MAOA algorithm is used to solve the spherical mining spanning tree (sphere MST) and communication network problems. Experimental results show that the proposed MAOA has achieved excellent results in terms of accuracy, robustness, and convergence speed.

Keywords

mathematical operator arithmetic optimization algorithm mining spanning tree spherical MST communication network problem meta-heuristic

1. Introduction

At present, with the continuous progress of natural science, the difficulty and depth of optimization problems in modern industrial are increasing. Optimization problems exist in many engineering fields, and have become one of the hotspots in the fields of medicine, engineering design and transportation (Ananduta et al., 2021). When faced with NP-hard problems in the field of engineering, traditional algorithms are usually unable to solve or the running time is unacceptable (Du & Leung, 1990). Considering these problems, meta-heuristic algorithms are widely used in many fields. Generally, optimization techniques are classified into two categories, namely meta-heuristic and deterministic algorithms. Because there’s nothing random about this strategy, giving the same input to a given problem always yields the same output (Hu et al., 2022). Deterministic algorithms may not be effective in solving multi-objective, large-scale, or multimodal problems. However, the optimization problem gradually tends to be complex in the real world. The structure of engineering optimization problems is more complex, and the parameter variables have continuous or discrete forms (Deng et al., 2023). Practical problems have larger scale, higher accuracy, and larger dimension, and the objective function is non-differentiable or nonlinear (Lima et al., 2021). Therefore, meta-heuristics with few parameters and strong adaptability provide a way to solve current optimization problems (Zhao et al., 2021) because of their simple and understandable principle, easy implementation, low computational cost, and strong robustness.

Meta-heuristic algorithms can be used for single objective or multi-objective optimization problems. They are divided into two important stages: exploration stage and development stage (Wang & Tan, 2017). Exploration behavior is a process in which individuals perform a global search in the solution space and find multiple sub-solution spaces. In order for more high-quality solutions to be found, the exploitation behavior is the process of population individuals searching for high-quality solutions in the adjacent neighborhood of the second-level solution space (Abualigah & Diabat, 2021). Exploration and exploitation behaviors are usually cooperative in order to approach the optimal solution (Gao et al., 2020). Four categories are typically used to categorize meta-heuristic algorithms.

(1)
Meta-heuristic algorithms based on evolution: Inspired by the evolution of species in nature, evolutionary algorithms mainly simulate the natural population evolution law to ensure that the proportion of high-quality individuals in the population is increasing. The earliest evolutionary algorithm is genetic algorithm (GA) (Deb et al., 2002), and other common evolutionary algorithms mainly include evolutionary strategy (ES) (Pincus, 1970), differential evolution algorithm (DEA) (Storn & Price, 1997), biogeography based optimization algorithm (BBOA) (Simon, 2008), evolutionary programming (EP) (Fogel, 1998), and immune algorithm (IA) (Wang et al., 2000).
(2)
Meta-heuristic algorithm based on swarm intelligence: Inspired by the collective behaviors and attributes of animals in nature, swarm intelligence algorithms mainly imitate the behavior of animals in hunting or other behaviors. Inspired by the living habits of different species, a variety of swarm intelligence algorithms emerged, in which each swarm is a group of organisms, and individuals in the swarm cooperate with each other to complete complex tasks in the real world. The most representative algorithm is the particle swarm optimization (PSO) proposed by Kennedy and Eberhart (1995). Ant colony optimization (ACO) (Dorigo et al., 2006) and artificial bee colony (ABC) (Karaboga & Akay, 2009) are also popular intelligent algorithms in such algorithms. The ant colony algorithm, proposed in 2006 by Dorigo et al., is inspired by ants looking for the shortest path located between their nest and the food, and imitates the behavioral process of ants looking for food. ABC, which is largely inspired by the social behavior of bees seeking food, classifies bees into three categories: employed bees, onlookers and scouts. Other popular swarm intelligence algorithms in recent years include ant lion optimization (ALO) (Mirjalili, 2015), gray wolf optimization (GWO) (Mirjalili et al., 2014), squirrel optimization algorithm (SSA) (Jain et al., 2019), whale optimization algorithm (WOA) (Mirjalili & Lewis, 2016), emperor penguin optimization algorithm (EPO) (Dhiman & Kumar, 2018), artificial hummingbird algorithm (AHA) (Zhao et al., 2022), chimp optimization algorithm (ChOA) (Khishe & Mosavi, 2020), firefly algorithm (FA) (Yang & Slowik, 2020), chicken swarm optimization algorithm (CSOA) (Meng et al., 2014), and marine predator algorithm (MPA) (Faramarzi et al., 2020).
(3)
Physics-based meta-heuristic algorithms are proposed inspired by physical phenomena or laws in nature. In recent years, widely recognized by the academic community include simulated annealing algorithm (SAA) (van Laarhoven & Aarts, 1987), gravitational search algorithm (GSA) (Rashedi et al., 2009), golden ratio optimization algorithm (GROA) (Nematollahi et al., 2020), black hole algorithm (BHA) (Hatamlou, 2013), sine cosine algorithm (SCA) (Mirjalili, 2016). In addition, the equilibrium optimizer (EO) (Faramarzi et al., 2020) was proposed inspired by the controlled volume mass balance. Artificial electric field algorithm (AEFA) (Anita, 2019) is proposed inspired by Coulomb’s law of electrostatic force; Henley gas solubility optimizer (HGSO) (Hashim et al., 2019) based on Henley’s Law, and so on, and they have been used by academics in various engineering problems.
(4)
Meta-heuristics based on human behavior that are currently widely recognized by the academic community include social learning optimization algorithm (SLOA) (Liu et al., 2016), volleyball premier league algorithm (VPLA) (Moghdani & Salimifard, 2018), teaching and learning optimization algorithm (TLBO) (Rao et al., 2012), election algorithm (EA) (Hojjata & Farnaz, 2015), brain storm optimization (BSO) (Shi, 2011), cultural evolution algorithm (CEA) (Kuo & Lin, 2013), and so on. Moghdani et al. (2020) proposed a multi-objective volleyball premier league algorithm, which has been applied to classical engineering problems. Kanwal et al. (2021) developed an improved election optimization algorithm which has been applied to an Internet of Things cloud data processing. Xu et al. (2022) proposed an improved instructional optimization algorithm which has been applied to remote fitness learning strategies.
Several representative meta-heuristics are shown in each category. AOA is widely popular due to its unique arithmetic properties. Short running time, simple principle and simple structure are the advantages of AOA (Abualigah et al., 2021). Because of the above advantages, Many engineering field problems have been solved using AOA, such as: optimizing deep neuro-fuzzy classifier (Talpur et al., 2022), optimizing deep convolutional spiking neural network model (Rajagopal et al., 2023), design problems of energy storage system in power grid (Kharrich et al., 2022), optimal power flow problem in DC networks (Montano et al., 2022), multi-objective problems in iot-enabled smart home (Bahmanyar et al., 2022), efficient text document clustering approach (Abualigah et al., 2022), and so on. But the no free lunch theorem states that no algorithm is applicable to all fields. Therefore, researchers improve the existing algorithms or propose new meta-heuristic algorithms according to the characteristics of the current problems (Zhong et al., 2021).

Tree is a common data structure in real life. It is widely used and come from graph theory (Diestel, 1997). In addition, the MST problem is often encountered in the field of power line network layout to obtain the most economical layout (Pop et al., 2018). This was first considered in 1926 by Boruvka (Nesetril & Nesetrilova, 2012). The famousness and importance of the MST originates from several aspects underlying: when faced with large graphs, efficient solutions exist, algorithms for solving the MST problem were proposed by Kruskal (1956), Prim (1957), Dijkstra (1959) and Sollin (1961). This problem has been applied in real-life and is widely used in engineering fields such as traffic problems, drainage system design, telecommunication network design, and distribution systems. Traditional methods used to solve MST problems are greedy strategies, which usually take a lot of time and can only obtain a MST. However, in most current problems, it is generally necessary to find a set of best or second-best values in a relatively short time. So, it is still necessary to find a method to solve the MST problem (Zhang et al., 2022). With the development of industry, the difficulty and scale of various problems continue to increase. In the real world, the earth that human beings live on is a sphere, so the research on the sphere becomes very practical significance. This paper further proves the effectiveness of the meta-heuristic algorithm and the sphere research in solving practical problems by studying the problem of laying optical cable in the world’s popular cities.

This article, to effectively reinforce the exploration and exploitation of arithmetic optimization algorithm (AOA) and reasonably achieve their balance, as well as the main contributions and motives for this study:
A novel mathematical operator-based arithmetic optimization algorithm is proposed.

The mathematical symmetry operator and mathematical median operator have been proposed and to improve the exploitation and exploration ability of the population, respectively.

The sine–cosine operator to effectively reinforce the exploration and exploitation of AOA algorithms and reasonably achieve their balance.

The MAOA algorithm is used to solve the spherical MST and communication network problems.

Experimental results show that the proposed MAOA has achieved excellent results in terms of global performance, accuracy, robustness, and convergence speed for solving the spherical MST problem.
The main contents of the other sections of this article are as follows: the related research work and spherical MST mathematical model In Section 2. In Section 3, arithmetic optimization algorithms are introduced and an improved arithmetic optimization algorithm for solving the spherical MST model is introduced. Section 4 experimental results are discussed and analyzed. In Section 5, examples of cable laying problems in popular cities around the world with different problem sizes are used to verify the performance of MAOA. And discusses the experimental results through fitness curves and variances, and so on. Section 6 is the summary and prospect of the future work.
2. Related Works

This section introduces two types of work related to spherical minimum spanning trees: minimum spanning tree problem, spherical geometry problem and spherical MST model. In addition, MST is a classic issue in graph theory, with important applications in the laying of optical cables between cities, the construction of highways, and the laying of water and gas pipelines. And it is widely recognized that meta-heuristic algorithms are employed to solve such problems. In the real world, the earth that human beings live on is a sphere, so the research on the sphere becomes very practical significance. This paper further proves the effectiveness of the meta-heuristic algorithm and the sphere research in solving practical problems by studying the problem of laying optical cable in the world’s popular cities.

2.1. MST Problem

The MST problem has been intensively studied in the last decades. However, it is more challenging and attractive for most researchers to solve more complex MST, such as the MST problem with conflict constraints and its variations considered by Zhang et al. (2011), the marked MST problem described by Consoli et al. (2013), the lower bounds and exact algorithms for the quadratic MST problem described by Pereira et al. (2015), the min-degree constrained MST problem with fixed centrals and terminals introduced by Dias et al. (2017), the angular constrained MST problem described by da Cunha and Lucena (2019), the capacitated MST problem with arc time windows presented by Kritikos and Ioannou (2021), the spherical MST problem presented by Bi et al. (2022) and Zhang et al. (2022) However traditional algorithms cannot effectively solve these problems at present. Next, this study proposes an improved AOA that uses mathematical methods: median operators, symmetric operators, and sine–cosine operators to better find solutions to the MST. And it extends the problem to the sphere by applying fiber optic cables to major cities around the world.

2.2. Spherical Geometry

The principle of spherical geometry: first fix a point, and then the set of all points r away from the point is a sphere. Where R needs to be greater than or equal to 0 and is called the radius of the sphere. And the center of the sphere is this fixed point, and does not belong to the sphere. $r = 1$ is a special case of the ball, called the unit ball. In analytic geometry, the set of all points $(X, Y, Z)$ with radius R is denoted by Equation (1).

X^{2} + Y^{2} + Z^{2} = r^{2}

(1)

A plane is used to intercept a sphere, and the resulting intersection is a circle. The area of the obtained tangent circle is maximized when the tangent plane passes through the center of the sphere. But the circle cut on the center of the sphere is called the small circle (Eldem & Ülker, 2017). An arc smaller than a semicircle is called a lower arc. The shortest distance between two different cities on Earth is through the center of the Earth and the lower arc of the tangent circle of these two cities (Uğur et al., 2009).

2.2.1. Representation of Points on the Sphere

Euclidean curves are one-dimensional objects whose positions along the path of three-dimensional curves can be described by t-parameters. Generally speaking, a vector function can represent any two points on a curve in Cartesian coordinates (Hearn, 1997).

P (t) = (x (t), y (t), z (t))

(2)

In normal circumstances, the coordinate equation can be shown by the parameter t, where t is between 0 and 1. For instance, the parametric form given by Equation (3) allows the definition of a circle with (0, 0) as its center in the x, y plane (Hearn, 1997).

{\begin{cases} x (t) = r \cos (2 π t) \\ y (t) = r \sin (2 π t) & 0 ⩽ t ⩽ 1 \\ z (t) = 0 \end{cases}

(3)

Any point on the sphere can be defined by Equation (4) parametric forms. An arbitrary curve on a three-dimensional space surface is a 2D variant represented by the parameters m and n. The implicit representation of a surface is given, that is, the algebraic equation contains three variables

x

y

, and

z

. Find three rational parametric equations, each of which contains m and n parameters.

P (u) = (x (m, n), y (m, n), z (m, n))

(4)

The three variables in Cartesian coordinates are represented by the parameters m and n, and these two parameters m and n are between 0 and 1, respectively. Set the origin to the center of the sphere and set the radius of the sphere to r, the spherical coordinate equation is given by Equation (5).

{\begin{cases} x (m, n) = r \cos (2 π m) \sin (π n) \\ y (m, n) = r \sin (2 π m) \sin (π n) & 0 ⩽ m ⩽ 1, 0 ⩽ n ⩽ 1 \\ z (m, n) = r \cos (π n) \end{cases}

(5)

Where the parameter m denoted the lines of longitude, and the parameter n denoted the lines of latitude on the spherical, as shown in Figures 1 to 3. The parameters m and n can represent all points on the spherical surface, and the parameters m and n correspond uniquely to the position coordinates of the city. In this article, the unit sphere is studied. Suppose that real-world engineering problems are not unit spheres, the value of coordinate x, y, and z are also expanded by the same factor.

Figure 1.

Meta-heuristic algorithms.

Figure 2.

Sphere.

Figure 3.

Lines of longitude and latitude.

2.2.2. The Geodesic of a Sphere

The arc of a section through the center of a sphere intersecting the surface of a sphere is a great circle. Geodesic usually refers to the shortest distance between different cities on the Earth (Lomnitz, 1995). The semi-circular arc connecting the north and south poles on the Earth’s surface is called the longitude. Each line of longitude has its own numerical value, called longitude. The orbit of a point on the Earth’s surface as the Earth rotates is called a latitude line, and any latitude line is circular and parallel in pairs. The equator is the largest, and the closer the circle gets to the poles, the smaller the circle.

Suppose there are two points $(P_{i}, P_{j})$ on the sphere. The geodetic line is shown in Figures 4 and 5. First of all, the vectors formed by the center of the sphere and the two points are $\vec{V_{i}} = (x_{i}, y_{i}, z_{i})$ and $\vec{V_{j}} = (x_{j}, y_{j}, z_{j})$ , and the angle between them is represented by theta( $θ$ ). Secondly, the scalar product between them is expressed by Equation (6).

\begin{aligned} \vec{V_{i}} \cdot \vec{V_{j}} & = ∣ \vec{V_{i}} ∣ ∣ \vec{V_{j}} ∣ c o s (θ) \end{aligned}

(6)

\begin{aligned} \vec{V_{i}} \cdot \vec{V_{j}} & = x_{i} x_{j} + y_{i} y_{j} + z_{i} z_{j} \end{aligned}

(7)

Figure 4.

The coordinates of the different positions.

Figure 5.

Geodesic lines for city $P_{i}$ and $P_{j}$ .

Finally, $θ$ and $\hat{d_{p 1, p 2}}$ are calculated using the following formula.

\begin{aligned} θ & = \arccos (\frac{x_{i} x_{j} + y_{i} y_{j} + z_{i} z_{j}}{r^{2}}) \end{aligned}

(8)

\begin{aligned} \hat{d_{p 1, p 2}} & = r \times θ = r \times \arccos (\frac{x_{i} x_{j} + y_{i} y_{j} + z_{i} z_{j}}{r^{2}}) \end{aligned}

(9)

In any two points on the sphere, the length from city

P_{i}

P_{j}

is equal to the geodesic from

P_{i}

P_{j}

. Suppose the problem size is n, the n-dimensional matrix is obtained by calculating the geodesic lines between different cities of the sphere. This matrix is represented as follows:

D = [\begin{matrix} \hat{n_{11}} & \hat{n_{12}} & \dots & \hat{n_{1 n}} \\ \hat{n_{21}} & \hat{n_{12}} & \dots & \hat{n_{2 n}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \hat{n_{n 1}} & \hat{n_{n 2}} & \dots & \hat{n_{n n}} \end{matrix}] [\begin{matrix} \infty & \hat{n_{12}} & \dots & \hat{n_{1 n}} \\ \hat{n_{21}} & \infty & \dots & \hat{n_{2 n}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \hat{n_{n 1}} & \hat{n_{n 2}} & \dots & \infty \end{matrix}]

(10)

Where

\hat{n_{i, j}}

denotes geodesic distance between city

P_{i}

and

P_{j}

. Where

\hat{n_{i, j}} = \infty

means that point

P_{i}

cannot to choose a route to reach itself.

2.3. Spherical MST Mathematical Model

An undirected graph can be represented by G (V, E), where V is a nonempty set for the 2D MST problem, called the vertex set. E is the set of unordered binary groups formed by the elements of V, called edge sets. The set of nodes, V, is a one-dimensional array, and the set of relationships between cities, E, is a two-dimensional data set, also known as the adjacency matrix. An edge between any two nodes has a corresponding weight w. x is a two-dimensional matrix, where $x_{i j} = 0$ or 1. If $X_{i j} = 1$ representing the edge between cities i and j is selected, otherwise, it is not selected. The spherical MST problem can be modeled as follows:

\begin{aligned} m i n f (x) = \sum_{i - 1}^{n - 1} \sum_{j = i + 1}^{n} w_{i j} x_{i j} \end{aligned}

(11)

\begin{aligned} s . t . \sum_{i - 1}^{n - 1} \sum_{j = i + 1}^{n} = n - 1 \end{aligned}

(12)

\begin{aligned} \sum_{v i \in S} \sum_{v j \in S, i < j} x_{i j} ⩽ ∣ S ∣ - 1, \forall S \subset V, ∣ S ∣ ⩾ 2 \end{aligned}

(13)

\begin{aligned} x_{i j} \in {0, 1}, {i, j} = x, y, \dots, n \end{aligned}

(14)

Equation (12) limits the final result to a tree structure. Moreover, in order to ensure that the solution set during the operation of the algorithm does not generate a closed loop, Equation (13) is added as a constraint.

For the 3D spherical MST problem, the node $P = {P_{1}, P_{2}, \dots, P_{n}}$ is a set of points on a sphere, where n is the problem size. Each node is denoted by $P_{i} = (x_{i}, y_{i}, z_{i})$ . The set of geodesic data for all nodes on the sphere is denoted by $A = {a_{i j} ∣ p_{i}, p_{j} \in V}$ . Calculate the lengths $\hat{d_{i, j}}$ of all geodesics by Equation (9). Equation (10) can be used to construct the $n \times n$ two-dimensional symmetric distance matrix D, where xij=0 or 1, represents selected and unselected. Finally, the mathematical model with Equation (15) as spherical MST is obtained

m i n f (x) = \sum_{i - 1}^{n - 1} \sum_{j = i + 1}^{n} \hat{d_{i j} x_{i j}}

(15)

Similarly, constraints 12–14 are applied to Equation (15).

3. Our Proposed Method

3.1. Spherical MST Mathematical Model

AOA is a meta-heuristic optimization algorithm based on population recently proposed by Abualigah et al. (2021). It is mainly based on the background of arithmetic operators in mathematics. Due to the properties of meta-heuristics, this algorithm can solve the optimization problem without gradient information, and the operation process of AOA first goes through initialization parameters and then optimization through exploration and exploitation stages.

3.1.1. Inspiration

The entire area of mathematics relies heavily on arithmetic. And usually number theory also provides some basic support for solving problems in real life. AOA is designed to take into account and is inspired by the rules of arithmetic operators in the field of mathematics.

3.1.2. Initialization Phase

The matrix X represents the initialization population of the algorithm, which are randomly generated. As the AOA continues to operate, the information of the population matrix continues to improve. The algorithm also records the best solution during the run.

X = [\begin{matrix} x_{1, 1} & \dots & \dots & x_{1, j} & \dots & x_{1, n} \\ x_{2, 1} & \dots & \dots & x_{2, j} & \dots & x_{2, n} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ x_{N - 1, 1} & \dots & \dots & x_{N - 1, j} & \dots & x_{N - 1, n} \\ x_{N, 1} & \dots & \dots & x_{N, j} & \dots & x_{N, n} \end{matrix}]

(16)

The exploitation behavior or the exploration behavior should be chosen during each run of the AOA. Therefore the math optimizer accelerated (MOA) is an important coefficient in AOA, and its iterative formula is shown in Equation (17).

MOA (CurrentIter) = Min + CurrentIter \times (\frac{Max - Min}{MaxIter})

(17)

Where CurrentIter represents the iteration number and is between 1 and MaxIter. Min and Max denote the upper and lower bounds during the iteration. The formulation of the math optimizer probability (MOP) in AOA.

MOP (CurrentIter) = 1 - (\frac{{CurrentIter}^{1 / α}}{{MaxIter}^{1 / α}})

(18)

Where CurrentIter indicate the number of iterations and MaxIter represents the maximum number of iterations of the algorithm.

α

is a constant parameter that determines the precision of exploitation. Abualigah et al. mention setting it to 0.499 in the article. (Abualigah et al., 2021).

3.1.3. Exploration Phase

The exploratory behavior of the AOA is introduced in this section. Depending on the operators in number theory, calculations using the multiplication (M) or division (D) operator can obtain lots of highly concentrated sub-optimal solutions or some optimal solutions. However, compared to the subtraction (S) and addition (A) operators, the M and D operators are more difficult to find the optimal solution objective due to their high dispersion. AOA conducts continuous random search in the solution space through D and M main search strategies.

The two major operators for AOA to find better solutions in the exploration stage are D and M, which are modeled in Equation (19). And the secondary phase is controlled by the accelerated math optimizer (MOA). If the value of MOA is less than the random number $r_{1}$ ( $r_{1}$ is a random number), AOA enters this stage. Next, the comparison with 0.5 is made by generating a random number $r_{2}$ between 0 and 1. Decide whether to choose the D or M operator.

x_{i, j} (CurrentIter) = {\begin{cases} best (x_{j}) \div (MOP + ℓ) \times (({UB}_{j} - {LB}_{j}) \times μ + {LB}_{j}) & r_{2} < 0.5 \\ best (x_{j}) \times MOP \times (({UB}_{j} - {LB}_{j}) \times μ + {LB}_{j}) & otherwise \end{cases}

(19)

Where

x_{i, j} (CurrentIter)

indicate the jth position of the ith solution in the next iteration, and

best (x_{j})

indicate that the jth position has so far been the best-obtained solution.

ℓ

indicate a quite small value. The algorithm’s upper and lower bounds are indicated by the letters UB and LB, respectively. In the studies of Abualigah et al.,

μ

is set to 0.5 (Abualigah et al., 2021).

3.1.4. Exploitation Phase

This section introduces AOA’s exploitation strategy. Calculations utilizing the subtraction (S) or addition (A) operators result in highly packed data. However, as opposed to other operators (D and M), Because of their minimal dispersion, these operators (S and A) may readily approach the target. In AOA, Based on these two primary search techniques, S and A operators randomly use the search space to discover the nearly ideal solution. Furthermore, the exploitation operators worked to help the exploitation stage by improving communication between them during the exploitation phase. The value of MOA determines whether the development phase is selected or not in AOA.

x_{i, j} (CurrentIter) = {\begin{cases} best (x_{j}) - MOP \times (({UB}_{j} - {LB}_{j}) \times μ + {LB}_{j}) & r_{3} < 0.5 \\ best (x_{j}) + MOP \times (({UB}_{j} - {LB}_{j}) \times μ + {LB}_{j}) & otherwise \end{cases}

(20)

Based on this equation, the first operator (S) is conditioned by

r_{3} < 0.5

(

r_{3}

is a random number). When

r_{3}

is greater than 0.5, the A is neglected. Otherwise, the A operator will be used for position update and the S operator will become invalid.

3.2. The MAOA Algorithm

In this section, an optimization algorithm called AOA based on mathematical strategy is introduced. It is used to solve the spherical MST problem with different problem sizes. For the traditional AOA position update based on the historical optimal individual, the population diversity will not be sensitive to the performance of the algorithm, and the development stage is easy to fall into local optimal. The position update of AOA is affected by upper and lower bounds, and the stride length is too large in some applications. MAOA avoids these shortcomings by introducing sine–cosine operators and mathematical median operators. First of all, adding sine–cosine operators to form vector triangles between population individual position and historical optimal individual position can improve AOA’s ability to jump out of local optimal. Secondly, the convergence accuracy of AOA can be effectively improved by introducing the median operator in the later stage. Finally, mathematical symmetry operators are introduced to help balance AOA exploration and development behavior. MAOA consists of traditional AOA, sine–cosine operators, mathematical median operators, and mathematical symmetry operators.

3.2.1. Sine–Cosine Operators

The update of AOA particle position was added, subtracted, multiplied and divided by the historical optimal individual. However, the population diversity could still play a role, and the exploration ability needed to be provided. In order to solve this problem, the sine-cosine operator is introduced, that is, AOA can be iteratively optimized through a variety of mathematical formulas. The following is the position update formula:

x_{i} (CurrentIter + 1) = X_{i} + \sin (r \times 2 π) (r + \cos (r \times 2 π) (best (X_{i}) - X_{i}))

(21)

Where

x_{i} (CurrentIter + 1)

indicate the position of the solution in the following iteration, and

best (x)

denotes the best solution obtained at the present time. r denotes a random number between 0 and 1,

x_{i}

is the position of the ith individual.

3.2.2. Mathematical Median Operator

In the operation of MAOA algorithm, a new operator is introduced on the basis of the original one by means of the median operator. Moreover, this operator can improve the convergence accuracy of the original algorithm and retain the excellent characteristics of AOA. This operator is inspired by the idea of finding the median between two points in mathematics and is used in AOAs. The updated formula is as follows:

x_{i} (CurrentIter + 1) = \frac{(best (X_{i}) + X_{i})}{2}

(22)

Where

x_{i} (CurrentIter + 1)

indicate the position of the solution in the following iteration, and best(X) denotes the best solution obtained at the present time.

x_{i}

is the position of the ith individual.

3.2.3. Mathematical Symmetric Operator

In geometric mathematics, there is an idea of symmetry which plays an important role. In mathematics, point A and point B are symmetric about the origin, that is, line segment AB passes through the origin, and the distance between point A and the origin is numerically equal to the distance between point B and the origin. In AOA, A mathematical symmetry operator can improve the algorithm’s capacity for exploration and is useful for striking a balance between development and exploration. The updated formula is as follows:

x_{i} (CurrentIter + 1) = best (X_{i}) + r \times (best (X_{i}) - X_{i})

(23)

Where

x_{i} (CurrentIter + 1)

indicate the position of the solution in the following iteration, and best(x) denotes the best solution obtained at the present time.

X_{i}

is the position of the ith individual. r represents a random number between 0 and 1.

3.3. Decoding and Encoding

3.3.1. MST Based on Prüfer Coding

Prüfer coding is the way to mark rootless trees. Its idea comes from Cayley’s theorem, which indicates that in a complete graph with n nodes, there exist $n^{n - 2}$ different trees. In combinational mathematics, Prüfer coding is a sequence transformed from a tree with labeled vertices, and a tree with n points yields a Prüfer sequence of length n-2. Of course, it can also be understood as a bijection between the spanning tree of the complete graph and the sequence. A Prüfer sequence corresponds to a unique unrooted tree, and an unrooted tree corresponds to a unique Prüfer sequence. Prüfer is not generally used to maintain tree structures and is often used for combinatorial counting problems.

3.3.2. Generation of Prüfer Sequence

Generating the Prüfer coding is an iterative process, assuming that a tree is constructed with seven nodes. The topology of this tree and the process of generating Prüfer sequence are shown in Figures 6 and 7. The following four steps make up the main portion of the process of transforming a tree into a Prüfer sequence:

(1)
Find the leaf node i on tree T that has the smallest value.
(2)
Suppose that the only node connected to node $i$ is $j$ , and then $j$ is the encoding of the first number, and the sequence of encoding from left to right.
(3)
The edges ( $i, j$ ) and node $i$ are removed, and a tree of n nodes is obtained.
(4)
This operation is repeated, and the algorithm is finished when there are two points left in the tree.

Figure 6.
Flowchart of the AOA.

Figure 7.
Prüfer sequence.

Through the above steps, the unique Prüfer sequence can be obtained, which is the permutation of $n - 2$ numbers between 1 and n.
3.3.3. The Generation of Spanning Trees is Based on Prüfer Sequence

According to the construction principle of Prüfer sequence and the primary transformation from Prüfer sequence to tree is mainly divided into the following four steps:

(1)
Finds the minimum node i that is not in the Prüfer sequence of the set, and store the node $n - 2$ at the end of the Prüfer sequence.
(2)
Connect this node with the first number of Prüfer sequence.
(3)
The node i in the set and the first number in Prüfer sequence are deleted.
(4)
This operation is repeated, and the algorithm is complete when there is no node in the Prüfer sequence.
Through the above steps, the unique spanning tree T of Prüfer sequence can be obtained.
3.3.4. Code Design

In this study, assuming that there are n nodes on the sphere, all nodes are numbered starting from 1. Thus, a tree of n nodes is uniquely represented by a Prüfer sequence of $n - 2$ integers. Then the dimension of each individual in the AOA is $n - 2$ . And the variables in the individual are rounded to obtain a sequence of integers. Suppose there is an individual is represented by

X_{i} : (3.11, 2.02, 3.21, 5.99, 2.22)

(24)

The Prüfer sequence obtained by rounding the

X_{i}

sequence is as follows:

X_{i} \to X_{j} : (3, 2, 3, 6, 2)

(25)

According to Prüfer sequence

X_{j}

, the spanning tree is as follows:

4. Experimental Results and Analysis

In order to verify the capability of MAOA for solving spherical MST problems, a variety of scale problems were used for benchmarking. The main contents of this section are as follows:

(1)
The problem scale and algorithm parameters are shown in the experimental setup section.
(2)
The experimental results of 25–400 dimensions are presented and compared in the small-scale section.
(3)
The experimental results of 500–1000 dimensions are presented and compared in the medium-scale section.
(4)
The experimental results of 1500–2000 dimensions are presented and compared in the large-scale section.

4.1. Experimental Setup

In this experiment, the unit sphere with n = 25, 50, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1500 and 2000 points (cities) is simulated respectively. All node data will not change during the run, and the same points will be used for all experiments. The experimental results in cities with a size less than or equal to 400 are from the literature (Bi et al., 2022), and between 500 and 1000 nodes of data and results from the literature (Zhang et al., 2022). In order to avoid the randomness of these algorithms, the intelligent algorithm involved in the experiment was run independently for 30 times.

MAOA is compared with the AOA, bio-inspired bare bones mayfly algorithm (BBMA), artificial electric field algorithm with inertia and repulsion (IRAEFA), PSO, GA, GWO, SMA (Li & Chen, 2020), DE, and ACO in best, worst, mean, standard deviation, and Friedman test. This paper effectively proves the performance of MAOA by compared the convergence curves, the ANOVA test, the average fitness values of 30 independent runs and the Wilcoxon rank-sum non-parametric statistical test.

In order to ensure the authenticity and reliability of the experimental results, MAOA will use the same system configuration as other algorithms. MATLAB 2018a was used to run the experiment, the processor on the device was Intel Core(TM) i7-9700U and the frequency was 3.00 GHz, the operating system was Windows10 and the RAM was 16 GB.

4.2. The Experimental Results of the Small Scale

In this section, the spherical MST is solved with 25, 50, 100, 200, 300, and 400 points (cities) of problem scale. The accuracy and performance of all algorithms are compared with other algorithms. The following results are obtained by running all algorithms for 30 times. Table 1 records the data results of these algorithms for 25–400 dimensions. Figures 8 and 9 displays the convergence curves in these four scenarios, Figure 10 shows the findings of the analysis of variance, Figure 11 shows the fitness values of the 30 runs, and Figure 12 shows the spherical MST for this experiment.

Figure 8.

Spanning tree.

Figure 9.

The convergence curves of algorithms for 25–400 points. (a) 25 cities; (b) 50 cities; (c) 100 cities; (d) 200 cities; (e) 300 cities; (f) 400 cities.

Figure 10.

The ANOVA test for 25–400 points. (a) 25 cities; (b) 50 cities; (c) 100 cities; (d) 200 cities; (e) 300 cities; (f) 400 cities.

Figure 11.

Fitness values for 25–400 points. (a) 25 cities; (b) 50 cities; (c) 100 cities; (d) 200 cities; (e) 300 cities; (f) 400 cities.

Figure 12.

The best route for 25–400 cities. (a) 25 cities; (b) 50 cities; (c) 100 cities; (d) 200 cities; (e) 300 cities; (f) 400 cities.

Table 1.

Experimental Result for the Ten Algorithms for 25, 50, $\dots$ , 400 Cities.

Points	Index	ACO	DE	PSO	GA	GWO	SMA	IRAEFA	BBMA	AOA	MAOA
25	Best	20.0309	16.9290	18.6661	22.7281	16.9782	15.0231	12.0511	13.6447	16.7977	13.4491
	Worst	24.0233	21.1576	25.1622	28.0316	27.0574	19.8995	16.6953	18.7544	21.9416	16.8748
	Mean	22.4738	18.8842	22.1693	26.1953	22.5108	17.6528	14.6396	15.8919	19.0534	14.7213
	Std	0.8548	1.0401	1.6687	1.2759	2.3995	1.2658	1.0752	1.0202	1.3235	0.9210
	p-value	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$2.12 \times 10^{- 6}$	0.2059	$2.37 \times 10^{- 5}$	$1.73 \times 10^{- 6}$	–
	p-value	1	1	1	1	1	1	0	1	1	–
50	Best	54.3648	48.4642	44.804	55.3518	51.8792	41.6473	27.6093	28.4447	49.3993	24.1250
	Worst	58.2811	54.5281	57.2689	64.8996	59.9504	54.8003	34.4254	37.2789	57.2610	31.5891
	Mean	56.7224	51.7636	51.506	61.6177	57.8262	47.6913	30.8955	34.817	52.5764	28.0005
	Std	1.0267	1.6024	3.0896	2.3682	1.5102	3.0873	1.8417	1.7546	2.1445	2.2032
	p-value	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.79 \times 10^{- 5}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	1	–
100	Best	125.2646	115.9142	105.388	122.6827	115.5757	102.8094	66.8479	69.2455	127.2740	52.3542
	Worst	131.6558	125.0204	122.9785	141.3354	132.9414	127.4766	79.0415	82.8067	136.2751	64.4948
	Mean	129.0522	120.9535	114.8626	136.5707	128.5117	116.7717	73.0467	76.9687	131.2906	57.3696
	Std	1.6687	2.0062	4.8750	3.5123	3.6147	4.8956	3.3247	3.358	2.6786	2.7674
	p-value	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	1	–
200	Best	269.3156	251.2678	241.6883	268.7868	266.3731	241.7519	166.8751	158.8478	271.0750	123.6745
	Worst	280.3753	265.8108	269.7025	296.2953	277.069	275.1255	197.8590	186.5355	284.4685	146.5015
	Mean	276.1739	261.2612	254.6991	284.4061	273.0821	260.9995	181.1215	175.9532	278.4517	136.9512
	Std	2.6751	3.3148	6.8041	5.834	2.7613	8.6126	7.6747	6.4658	3.4083	5.1802
	p-value	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.79 \times 10^{- 5}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	1	–
300	Best	404.8393	381.7567	390.3274	422.2974	416.0361	391.2315	289.9457	275.7485	412.3600	227.8733
	Worst	421.2182	410.6107	425.314	448.4215	425.4912	417.1096	328.7072	320.7405	428.0944	264.6494
	Mean	415.9465	402.4801	412.8644	437.9276	420.2078	404.5279	306.8270	298.7668	423.1081	244.4062
	Std	3.7998	5.3663	8.6242	5.782	2.5851	6.1692	11.2929	10.9117	4.2971	9.5564
	p-value	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.79 \times 10^{- 5}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	1	–
400	Best	559.3794	534.0305	535.9726	562.2246	558.2885	545.4407	422.8765	387.1640	552.2144	345.8756
	Worst	579.4299	561.5491	563.4369	601.2813	576.4934	577.0593	472.5275	438.9316	577.3100	428.3214
	Mean	572.5480	554.6098	552.9052	586.7195	568.7354	561.7902	443.1054	415.9261	568.6876	375.8165
	Std	5.0014	4.9569	7.1746	7.9634	4.3849	8.3622	13.3465	13.1962	5.5193	19.4988
	p-value	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	$1.79 \times 10^{- 5}$	$2.60 \times 10^{- 6}$	$1.73 \times 10^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	1	–

MAOA is compared with other algorithms in 25–400 cities respectively. The running results are shown in Table 1. Compared with IRAEA, the MAOA proposed in this study has lower convergence accuracy when the scale is 25 cities, but its stability is better than other algorithms. At the same time, the findings of the Friedman test, the best, worst, and mean values are superior to those of alternative methods. Among the 50 cities, the standard deviation of MAOA is 0.9210, which is lower than ACO, but the best, worst and average values are superior to other algorithms. When the number of cities increases to 100, 200, 300 and 400, MAOA is better than other algorithms in terms of best, mean values and Friedman, and its standard deviation is also better than most algorithms. The disadvantage is that its standard deviation is lower than ACO in some cases. As the scale of the problem continues to complexity, MAOA’s Friedman ranking continues to be the best. Table 1 shows the results of Wilcoxon rank sum test at low dimensions. The p value is very small, so it can be seen that MAOA is significantly superior to other algorithms.

Figure 10 shows the convergence curve of all algorithms. It can be seen from Figure 9 that when running for about 100 generations, it is better than most algorithms and still searching for the optimal solution. The optimization accuracy of DE, AOA and ACO algorithms is relatively low. It can be seen that when the number of cities is less than 400, the convergence accuracy and speed of MAOA achieve good results with the scale of the problem continues to complexity.

Figure 10 shows the variance graph of all algorithms. It can be seen that MAOA, ACO and DE have high stability in most cases, while GWO and PSO have poor stability. Figure 11 shows the optimal value of 30 runs. It can be clearly seen that MAOA is superior to other algorithms in most cases. When other algorithms become very slow to update, MAOA algorithm can still continue to search for optimization. With the scale of the problem continues to complexity, MAOA outperforms other algorithms. Figure 12 displays the spherical MST paths for 25–400 cities.

4.3. The Experimental Results of the Medium-scale

In this section, the proposed algorithm MAOA is simulated with other nine algorithms at 500–1000 cities respectively. Table 2 records the data results of these algorithms in 500–1000 dimensions. The use of bold denotes that it is the optimal solution for the nine algorithms. Figure 13 shows the convergence curves of the six problem sizes, Figure 14 shows the results of the analysis of variance of the six problem sizes, and Figure 15 shows the average fitness values of the 30 runs. And the spherical MST is listed in Figure 16.

Figure 13.

The convergence curves of algorithms for 500–1000 points. (a) 500 cities; (b) 600 cities; (c) 700 cities; (d) 800 cities; (e) 900 cities; (f) 1000 cities.

Figure 14.

The ANOVA test for 500–1000 points. (a) 500 cities; (b) 600 cities; (c) 700 cities; (d) 800 cities; (e) 900 cities; (f) 1000 cities.

Figure 15.

Fitness values for 500–1000 points. (a) 500 cities; (b) 600 cities; (c) 700 cities; (d) 800 cities; (e) 900 cities; (f) 1000 cities.

Figure 16.

The best route for 500–1000 cities. (a) 500 cities; (b) 600 cities; (c) 700 cities; (d) 800 cities; (e) 900 cities; (f) 1000 cities.

Table 2.

Experimental Result for the Ten Algorithms for 400,500, $\dots$ , 1000 Cities.

Points	Index	ACO	DE	PSO	GA	GWO	SMA	BBMA	AOA	MAOA
500	Best	712.2368	693.7499	700.4049	722.1846	710.3885	690.8498	526.7288	710.6879	470.3627
	Worst	728.7615	712.1170	727.0427	761.1239	730.1305	727.9512	576.1991	730.6225	540.7375
	Mean	721.7657	703.2983	714.2092	743.3494	721.7850	714.6237	551.4301	721.9112	505.7130
	Std	4.1074	4.7147	6.8842	10.3560	4.8781	8.2787	10.9296	5.7014	17.5145
	p-value	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.79 \times^{- 5}$	$1.73 \times^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	–
600	Best	845.7269	836.9045	830.4673	883.8227	859.0083	834.3700	661.5117	850.6166	624.0554
	Worst	874.7280	861.8101	875.6180	910.6815	881.7119	866.6941	720.9847	880.3411	676.8493
	Mean	865.3516	851.2513	856.0039	899.1887	871.8175	854.6719	687.1937	872.9360	647.8017
	Std	7.1528	6.0473	12.6292	6.4852	4.6854	9.0553	15.486	6.0097	17.1088
	p-value	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.79 \times^{- 5}$	$1.73 \times^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	–
700	Best	997.0200	996.3382	991.1713	1019.2579	1014.0621	991.4004	795.8841	1015.7392	730.6666
	Worst	1028.7226	1011.0239	1028.8977	1070.9165	1033.3789	1022.5838	887.1201	1042.4202	814.6608
	Mean	1017.4256	1003.0214	1009.3892	1047.1824	1024.7324	1010.1426	826.7429	1030.1895	771.2698
	Std	7.6259	4.1231	12.1345	12.7852	5.2124	7.9722	20.7736	6.6162	24.2979
	p-value	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.79 \times^{- 5}$	$1.73 \times^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	–
800	Best	1159.6173	1128.1320	1129.6123	1188.4263	1153.7342	1137.4899	920.5137	1131.0289	896.2293
	Worst	1185.3013	1163.4454	1175.3558	1224.0369	1185.4567	1185.2648	1003.667	1167.2801	973.0744
	Mean	1176.5963	1153.9216	1157.8010	1204.1616	1171.0356	1166.0980	961.2036	1159.4564	933.5114
	Std	6.5324	6.5457	13.8735	9.5860	7.5634	11.9602	21.599	7.5454	16.2081
	p-value	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.79 \times^{- 5}$	$1.73 \times^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	–
900	Best	1364.4897	1356.2774	1346.2868	1372.1236	1365.4940	1357.1972	1072.67	1369.3126	1228.4144
	Worst	1380.5913	1366.5276	1373.0494	1393.6464	1376.8187	1376.3410	1167.294	1380.7887	1257.3631
	Mean	1374.7512	1360.9010	1364.1205	1384.5569	1372.1504	1367.2020	1107.199	1376.5435	1241.4381
	Std	3.7688	2.8044	7.3119	6.2967	2.4914	5.1026	28.7825	3.2142	7.5412
	p-value	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.79 \times^{- 5}$	$1.73 \times^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	–
1000	Best	1476.3932	1452.4452	1443.5458	1500.2107	1464.5448	1454.3861	1167.5380	1481.3639	1170.4903
	Worst	1499.3696	1475.0927	1490.7540	1542.2565	1494.2965	1496.3472	1344.461	1511.6308	1260.2146
	Mean	1485.2603	1461.4399	1470.4062	1520.8039	1481.8861	1477.4963	1248.647	1498.4057	1213.0124
	Std	5.9182	6.0035	11.5211	11.1323	7.5045	10.0101	34.2288	9.1073	19.7603
	p-value	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.73 \times^{- 6}$	$1.79 \times^{- 5}$	$1.73 \times^{- 6}$	–
	p-value	1	1	1	1	1	1	1	1	–

Table 2 shows that in the 500 and 600 cities; different algorithms have significant differences in search ability. The table shows that even though MAOA’s standard deviation is not ideal, the best, worst, and average values all come out on top. In addition, Friedman has the worst GA ranking in the test and MAOA’s Friedman ranked best. Figure 13 shows the convergence curves of these algorithms in the 500 and 600 cities. Figure 14 shows the ANOVA results of each algorithm under 500 and 600 cities. The figure shows that the stable results of MAOA are also within the allowed range. Figures 15 and 16 respectively show the fitness values and MST of these algorithms run for 30 times in 500 and 600 cities.

Table 2 displays the simulation results of these algorithms in 700 and 800 cities. The table shows that MAOA performs well in best, worst, and average values, but the standard deviation is not satisfactory. The experiment shows that GA’s comprehensive performance is the worst. DE and ACO get the optimal value of standard deviation in the case of 700 and 800 cities respectively, and MAOA gets the optimal value in other cases. Taking all the metrics together, MAOA ranked the best in Friedman’s test. Figure 13 shows the convergence curves of these algorithms in 700 and 800 cities, and it is clear that MAOA has the fastest and most accurate convergence. Figure 14 shows the analysis of variance results of these algorithms in the case of 700 and 800 cities, and demonstrates that the MAOA stability results are also within the permitted range. Figures 15 and 16 show the fitness values and spherical MST of these algorithms run for 30 times in 700 and 800 cities, respectively.

The cases of 900 and 1000 cities are also shown in Table 2. At 900 cities, MAOA performed poorly in terms of best, worst and mean values, but the standard deviation of 7.5412 was better than BBMA’s 28.7825. When increasing to 1000 cities, MAOA outperforms other algorithms in terms of best, worst, mean values and standard deviation, and is also optimal in Friedman test. Figures 13 and 14 respectively show the analysis of convergence curves and variance results of these algorithms in the case of 900 and 1000 cities. It is evident that MAOA’s convergence speed and accuracy are at their best, and its stability findings are within acceptable bounds. Figures 15 and 16 show the fitness values and spherical MST of these algorithms run for 30 times in 900 and 1000 cities respectively.

4.4. The Experimental Results of the Large-scale

After the above comparison, MAOA showed good performance. However, in the face of the increasing complexity of problems in the real world, in this section, MAOA will be tested at a higher problem size, where the city size is 1500 and 2000. Table 3 records the best, worst, mean and standard deviation, Wilcoxon rank and nonparametric test results and mean ranking at 1500 and 2000 dimensions. The usage of bold signifies that it is the best of these algorithms. Figure 17 displays the convergence curves of the two problem sizes, and shows the results of the analysis of variance of the two problem sizes, and shows the average fitness values of the 30 runs. And the spherical MST is listed in Figure 18.

Figure 17.

The ANOVA test for 1500–2000 points. (a) 1500 cities; (b) 2000 cities; (c) 1500 cities; (d) 2000 cities; (e) 1500 cities; (f) 2000 cities.

Figure 18.

The best route for 1500–2000 cities. (a) 1500 cities; (b) 2000 cities.

Table 3.

Experimental Result for the Ten Algorithms for 1500 and 2000 Cities.

Points	Index	ACO	DE	PSO	GA	GWO	SMA	AOA	MAOA
1500	Best	2221.7795	2197.0312	2197.6713	2269.2320	2221.9422	2224.1298	2243.1303	1891.9541
	Worst	2260.3448	2229.5322	2258.1486	2316.8482	2262.7547	2256.9568	2277.2286	1985.5960
	Mean	2244.2728	2213.1582	2233.4391	2294.7583	2248.6032	2239.5721	2262.7632	1932.8051
	Std	8.9349	13.2616	19.5029	12.5906	8.8857	9.0869	11.3464	24.0827
	p-value	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	–
	p-value	1	1	1	1	1	1	1	–
2000	Best	2999.8031	2944.1027	2970.3788	3043.7567	2994.4882	2975.3836	2982.7120	2609.1130
	Worst	3033.1283	2999.6843	3025.2705	3090.3617	3032.1030	3028.9044	3038.1727	2711.1779
	Mean	3015.5001	2983.0037	3007.7928	3068.1324	3017.0602	3011.3795	3020.4140	2675.5730
	Std	8.2480	12.6730	14.6142	11.0223	8.9656	13.2468	10.0592	23.4745
	p-value	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	$1.734 \times 10^{- 6}$	–
	p-value	1	1	1	1	1	1	1	–

Table 3 shows the running results of MAOA and other algorithms in high dimensions of 1500 and 2000, respectively. The best, worst, average and standard values are compared respectively, and Wilcoxon rank and nonparametric test results are given. The table shows that MAOA outperforms other algorithms in terms of average, best, and worst values. For 1500 and 2000 dimensions, respectively, its standard deviation yields the best result. Table 3 displays the Wilcoxon rank sum test outcomes in high dimensions. The p value is very small, so it is clear that MAOA is vastly better than other algorithms. Obviously, MAOA performs superior in terms of convergence accuracy and speed, and its exploration capabilities are enhanced and contribute to the balance between exploration and exploitation. Figure 17 shows the results of variance analysis. AOA is more stable than MAOA. Figure 17 displays the fitness values for 30 independent runs. In addition, Figure 18 shows the MST of MAOA in different dimensions.

5. Communication Network Problem

The MST problem is an important problem in the design of communication networks. In order to further verify the efficiency and performance of the proposed MAOA algorithm, different algorithms are used to compare and analyze the engineering design optimization problems. It is proved that MAOA has strong ability in finding acceptability and fast convergence speed. In this test, the population individual of all meta-heuristics is set to 30, the iteration number is 300, and they are run independently 10 times.

Communication network is a common problem in real life, which can be solved by finding the lowest cost objective function. Usually the region involved is mapped to a two-dimensional plane to solve, but it gets bigger and bigger according to the complexity and scale of the problem. This experiment extends the problem to three-dimensional spheres and selects some popular cities on Earth, and the number of cities is divided into 100, 200 and 300 for experimental simulation. The latitude and longitude coordinate data of the cities in the experiment are all from Math Works MATLAB Mapping toolbox. The experimental results are shown in Figure 19. It can be observed that the optimal solution and average value sought by MAOA are better than other algorithms, and MAOA provides a high-quality reference plan for decision makers.

Figure 19.

The best route around the global for 100–300 cities. (a) 100 cities for eastern hemisphere; (b) 100 cities for western hemisphere; (c) 200 cities for eastern hemisphere; (d) 200 cities for western hemisphere; (e) 300 cities for eastern hemisphere; (f) 300 cities for western hemisphere.

It is clear from the simulation results data in Table 4 that MAOA exhibits strong search performance. By adding mathematical strategies, the best outcome in terms of the ideal solution and average value can be obtained with MAOA, which can swiftly converge to the global optimal solution. MAOA can achieve excellent performance from the following aspects.

Table 4.

Experimental Result for the Eight Algorithms for 100, 200 and 300 Cities.

Points	Index	ACO	DE	PSO	GA	GWO	SMA	AOA	MAOA
100	Best	$6.2799 \times 10^{5}$	$5.7411 \times 10^{5}$	$5.945 \times 10^{5}$	$6.7401 \times 10^{5}$	$6.0739 \times 10^{5}$	$4.7792 \times 10^{5}$	$6.2721 \times 10^{5}$	$2.3493 \times 10^{5}$
	Worst	$6.6607 \times 10^{5}$	$6.2169 \times 10^{5}$	$6.5078 \times 10^{5}$	$7.1674 \times 10^{5}$	$6.5038 \times 10^{5}$	$5.4724 \times 10^{5}$	$6.7848 \times 10^{5}$	$2.9790 \times 10^{5}$
	Mean	$6.4812 \times 10^{5}$	$6.0454 \times 10^{5}$	$6.2226 \times 10^{5}$	$6.9145 \times 10^{5}$	$6.3230 \times 10^{5}$	$5.1488 \times 10^{5}$	$6.4745 \times 10^{5}$	$2.6053 \times 10^{5}$
	Std	$1.4426 \times 10^{4}$	$1.2046 \times 10^{4}$	$1.6089 \times 10^{4}$	$1.4042 \times 10^{4}$	$1.1161 \times 10^{4}$	$2.1144 \times 10^{4}$	$1.6173 \times 10^{4}$	$1.7276 \times 10^{4}$
200	Best	$1.2880 \times 10^{6}$	$1.2540 \times 10^{6}$	$1.2771 \times 10^{6}$	$1.4657 \times 10^{6}$	$1.3405 \times 10^{6}$	$1.1356 \times 10^{6}$	$1.1690 \times 10^{6}$	$6.2735 \times 10^{5}$
	Worst	$1.3407 \times 10^{6}$	$1.3132 \times 10^{6}$	$1.3663 \times 10^{6}$	$1.5149 \times 10^{6}$	$1.3738 \times 10^{6}$	$1.1844 \times 10^{6}$	$1.2016 \times 10^{6}$	$7.3260 \times 10^{5}$
	Mean	$1.3204 \times 10^{6}$	$1.2812 \times 10^{6}$	$1.3289 \times 10^{6}$	$1.4871 \times 10^{6}$	$1.3608 \times 10^{6}$	$1.1613 \times 10^{6}$	$1.1881 \times 10^{6}$	$6.7193 \times 10^{5}$
	Std	$1.6239 \times 10^{4}$	$1.6221 \times 10^{4}$	$2.7477 \times 10^{4}$	$1.6661 \times 10^{4}$	$1.0394 \times 10^{4}$	$1.6038 \times 10^{4}$	$1.0333 \times 10^{4}$	$3.0049 \times 10^{4}$
300	Best	$2.1074 \times 10^{6}$	$2.0609 \times 10^{6}$	$2.0364 \times 10^{6}$	$2.2402 \times 10^{6}$	$2.1379 \times 10^{6}$	$1.7954 \times 10^{6}$	$2.0511 \times 10^{6}$	$1.2343 \times 10^{6}$
	Worst	$2.1738 \times 10^{6}$	$2.1260 \times 10^{6}$	$2.1729 \times 10^{6}$	$2.3243 \times 10^{6}$	$2.1982 \times 10^{6}$	$1.8492 \times 10^{6}$	$2.1090 \times 10^{6}$	$1.3239 \times 10^{6}$
	Mean	$2.1438 \times 10^{6}$	$2.0933 \times 10^{6}$	$2.1065 \times 10^{6}$	$2.2820 \times 10^{6}$	$2.1691 \times 10^{6}$	$1.8206 \times 10^{6}$	$2.0758 \times 10^{6}$	$1.2766 \times 10^{6}$
	Std	$2.1232 \times 10^{4}$	$2.2441 \times 10^{4}$	$4.3821 \times 10^{4}$	$2.7227 \times 10^{4}$	$2.1849 \times 10^{4}$	$1.7123 \times 10^{4}$	$2.0959 \times 10^{4}$	$2.7014 \times 10^{4}$

First of all, MAOA is consistent with the two-stage optimization strategy of AOA in design principle, on the one hand, finding sub-domains through global exploration, and on the other hand, deep optimization through exploitation in each sub-domain. With the operation of the algorithm, it gradually changes from exploration behavior to exploitation behavior, which improves the balance between exploration and exploitation. Secondly, sine–cosine operators can add a new optimization method for AOA, which can avoid the single mode of population in the optimization process. The mathematical symmetric operator keeps jumping out of the local behavior to avoid population deterioration and make the population develop better. The median operator can effectively help MAOA dynamically adjust the stride length in the development stage and can be used to improve the accuracy. This improves the probability of finding the global optimum and the convergence rate of MAOA.

Although the aforementioned experimental results demonstrate that MAOA performs better than most algorithms, it still has certain drawbacks and restrictions. When the problem scale is small, the speed and accuracy of MAOA are not much different from most algorithms, but worse than some improved algorithms. However, with the increase of problem scale, the accuracy of MAOA consistently achieves excellent results compared to other algorithms. For the global optical cable laying problem, it can be clearly seen that MAOA has a higher level of ability to find the best solution compared with the other seven algorithms. The main reason is that AOA is designed to solve these kinds of problems. Adding binary encoding enables MAOA to solve a wider variety of practical optimization problems.

6. Conclusions and Future Directions

AOA algorithm is a meta-heuristic optimization algorithm based on mathematical model. AOA has its simple and direct implementation, which is highly adaptable to the problems in real life. It does not require resizing the overall size and stopping many parameters beyond the criteria. Its built-in coefficients MOA and MOP also improve the balance of the AOA search process. It can be done without requiring a derivative. It can effectively solve many problems in continuous space. According to NFL theorem, AOA itself also has some shortcomings. The sine–cosine operators proposed in this paper can add new optimization operators to AOA, and increasing the diversity of optimization methods can avoid the fixed mode of population in the optimization process. Mathematical symmetry operator is a case of applying mathematical geometry method to AOA. It makes the individual of the population constantly jump out of the local behavior to avoid the population into the local optimal, so that the population can develop better. Median operator adopts the idea of median between any two points in mathematics, which can effectively help MAOA dynamically adjust stride length in the development stage and can be used to improve accuracy. In this paper, the spherical MST problem is solved on 14 different 3D spheres with BBMA, IRAEFA, GA, PSO, GWO, SMA, DE, ACO and AMO The performance of the proposed MAOA is verified by comprehensive experiments. Therefore, MAOA algorithm is a more suitable algorithm for solving spherical MST problem. According to NFL theorem, it is impossible for an algorithm to show superior performance for all optimization problems. Like all algorithms, MAOA has some drawbacks. MAOA currently has two major drawbacks: long running time and improved stability. Of course, MAOA has other drawbacks. These problems are also within the scope of practical application to some extent. Next, MAOA will be further studied to solve the spherical MST problem and solve the above two shortcomings. Finally, the improved algorithm will be applied to solve the spherical MST problem, such as eliminating the noise of the femur surface and directional position estimator. In addition, its practical difficulties in other engineering fields will also be considered for application, such as last-kilometer distribution, logistics center sit, three-dimensional path planning, thermoelectric dispatching and other problems.

Footnotes

Acknowledgements

The authors express great thanks to the financial support from the National Natural Science Foundation of China.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Authors’ Contribution

Q. Ldid supervision, writing-review and editing; X. M carried out the MAOA algorithm studies; Y.W participated in the drafted manuscript; Y.Z carried out the review and editing. All authors read and approved the final manuscript.

Consent for Publication

Not applicable.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by the National Natural Science Foundation of China (Grant numbers 62066005, U21A20464).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Availablity of Data and Materials

Not applicable.

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Arithmetic Optimization Algorithm With Mathematical Operator for Solving Spherical Minimum Spanning Tree Problem

Abstract

Keywords

1. Introduction

2.1. MST Problem

2.2. Spherical Geometry

3.1. Spherical MST Mathematical Model

3.1.1. Inspiration

3.1.2. Initialization Phase

3.2.1. Sine–Cosine Operators

3.3.1. MST Based on Prüfer Coding

3.3.2. Generation of Prüfer Sequence

4.2. The Experimental Results of the Small Scale

Footnotes

Acknowledgements

Ethical Approval

Authors’ Contribution

Consent for Publication

Funding

Declaration of Conflicting Interests

Availablity of Data and Materials

References